Modelling and forecasting mortality distributions in England and Wales using the Lee-Carter model

Lee and Carter proposed in 1992 a non-linear model m_xt = exp (a_x + b_x k_t + ε_xt) for fitting and forecasting age-specific mortality rates at age x and time t. For the model parameter estimation, they employed the singular value decomposition method to find a least squares solution. However, the singular value decomposition algorithm does not provide the standard errors of estimated parameters, making it impossible to assess the accuracy of model parameters. This article describes the Lee-Carter model and the technical procedures to fit and extrapolate this model. To estimate the precision of the parameter estimates of the Lee-Carter model, we propose a binomial framework, whose parameter point estimates can be obtained by the maximum likelihood approach and interval estimates by a bootstrap approach. This model is used to fit mortality data in England and Wales from 1951 to 1990 and to forecast mortality change from 1991 to 2020. The Lee-Carter model fits these mortality data very well with R² being 0.9980. The estimated overall age pattern of mortality a_x is very robust whereas there is considerable uncertainty in b_x (changes in the age pattern over time) and k_t (overall change in mortality). The fitted log age-specific mortality rates have been declining linearly from 1951 to 1990 at different paces and the projected rates will continue to decline in such a way in the 30 years prediction period.     

Lee–Carter mortality forecasting: a parallel generalized linear modelling approach for England and Wales mortality projections

Summary. The paper presents a reinterpretation of the model underpinning the Lee–Carter methodology for forecasting mortality (and other vital) rates. A parallel methodology based on generalized linear modelling is introduced. The use of residual plots is proposed for both methods to aid the assessment of the goodness of fit. The two methods are compared in terms of structure and assumptions. They are then compared through an analysis of the gender- and age-specific mortality rates for England and Wales over the period 1950–1998 and through a consideration of the forecasts generated by the two methods. The paper also compares different approaches to the forecasting of life expectancy and considers the effectiveness of the Coale–Guo method for extrapolating mortality rates to the oldest ages.     

1981 census evaluation programme in England and Wales

The program developed to evaluate the coverage and quality of results from the 1981 census of England and Wales is described. "A post-enumeration survey was the main tool used, but for the evaluation of census coverage, this was augmented by a series of demographic checks against statistics from other administrative sources. The main conclusion from the coverage checks was that the census probably missed about 241,000 people net (about half of one per cent of the population) including some 36,000 children aged 0-4. At older ages than this, adults aged 16-44 were more likely to be missed than others and males rather more than females. Students and people out of employment were also more likely to be missed than people in employment. The quality of householders' responses to particular census questions was evaluated in a detailed post-enumeration interview survey. The results of this showed that the questions subject to most response error were those on rooms, various aspects of economic activity and the main means of travel to work."     

Updating small area population estimates in England and Wales

"Population estimates have important implications for resource allocation within government and commerce, and are often assumed to be without error. Currently, central government provides annual population estimates for all the local and health authority districts in Britain, but estimates are needed for smaller areas, typically for electoral wards and postal sectors. Small area estimates are provided by some local authorities and commercial organizations, using different methods; the accuracy of these estimates is modelled here within a multilevel framework. Certain characteristics of the small area and of the method of estimation are included as explanatory variables. Results show that the method of estimation used is of great importance."     

Trends in inequality in infant mortality in the north of England, 1921–1973, and their association with urban and social structure

Summary.  The paper analyses a time series of infant mortality rates in the north of England from 1921 to the early 1970s at a spatial scale that is more disaggregated than in previous studies of infant mortality trends in this period. The paper describes regression methods to obtain mortality gradients over socioeconomic indicators from the censuses of 1931, 1951, 1961 and 1971 and to assess whether there is any evidence for widening spatial inequalities in infant mortality outcomes against a background of an overall reduction in the infant mortality rate. Changes in the degree of inequality are also formally assessed by inequality measures such as the Gini and Theil indices, for which sampling densities are obtained and significant changes assessed. The analysis concerns a relatively infrequent outcome (especially towards the end of the period that is considered) and a high proportion of districts with small populations, so necessitating the use of appropriate methods for deriving indices of inequality and for regression modelling.     

Quantifying uncertainty in the biospheric carbon flux for England and Wales

Summary.  A crucial issue in the current global warming debate is the effect of vegetation and soils on carbon dioxide (CO₂) concentrations in the atmosphere. Vegetation can extract CO₂ through photosynthesis, but respiration, decay of soil organic matter and disturbance effects such as fire return it to the atmosphere. The balance of these processes is the net carbon flux. To estimate the biospheric carbon flux for England and Wales, we address the statistical problem of inference for the sum of multiple outputs from a complex deterministic computer code whose input parameters are uncertain. The code is a process model which simulates the carbon dynamics of vegetation and soils, including the amount of carbon that is stored as a result of photosynthesis and the amount that is returned to the atmosphere through respiration. The aggregation of outputs corresponding to multiple sites and types of vegetation in a region gives an estimate of the total carbon flux for that region over a period of time. Expert prior opinions are elicited for marginal uncertainty about the relevant input parameters and for correlations of inputs between sites. A Gaussian process model is used to build emulators of the multiple code outputs and Bayesian uncertainty analysis is then used to propagate uncertainty in the input parameters through to uncertainty on the aggregated output. Numerical results are presented for England and Wales in the year 2000. It is estimated that vegetation and soils in England and Wales constituted a net sink of 7.55 Mt C (1 Mt C = 10¹² g of carbon) in 2000, with standard deviation 0.56 Mt C resulting from the sources of uncertainty that are considered.     

Infant mortality rates: time trends and fractional integration

This paper examines the existence of time trends in the infant mortality rates in a number of countries in the twentieth century. We test for the presence of deterministic trends by adopting a linear model for the log-transformed data. Instead of assuming that the error term is a stationary I(0), or alternatively, a non-stationary I(1) process, we allow for the possibility of fractional integration and hence for a much greater degree of flexibility in the dynamic specification of the series. Indeed, once the linear trend is removed, all series appear to be I(d) with 0<d<1, implying long-range dependence. As expected, the time trend coefficients are significantly negative, although of a different magnitude from those obtained assuming integer orders of differentiation.     

Violent crime and incentives in the long-run: evidence from England and Wales

This study uses recent advances in time-series econometrics to investigate the non-stationarity and co-integration properties of violent crime series in England and Wales. In particular, we estimate the long-run impact of economic conditions, beer consumption and various deterrents on different categories of recorded violent crime. The results suggest that a long-run causal model exists for only minor crimes of violence, with beer consumption being a predominant factor.     

Exploring the economic and social determinants of psychological well-being and perceived social support in England

Summary.  A fundamental focus of Government concern is to enhance well-being. Recently, policy makers in the UK and elsewhere have recognized the importance of the community and society to the well-being of the nation as a whole. We explore the extent to which economic and social factors influence the psychological well-being of individuals and their perceptions of the social support that they receive, using Health Survey for England data. We employ a random-effects ordered probit modelling approach and find that unobserved intrahousehold characteristics help to explain the variation in our dependent variables, particularly for co-resident females. Our results indicate that individuals with acute and chronic physical illness, who are female, unemployed or inactive in the labour market and who live in poor households or areas of multiple deprivation report lower levels of psychological well-being. Reduced perceptions of social support are associated with being male, single or post marriage, from an ethnic minority, having low educational attainment and living in a poor household, but are not statistically related to area deprivation measures. These findings may help to inform the contemporary policy debate surrounding the promotion of individual well-being and community, through the alleviation of social exclusion.     

Combining census and registration data to estimate detailed elderly migration flows in England and Wales

Summary.  A log-linear model is developed to estimate detailed elderly migration flows by combining data from the 2001 UK census and National Health Services patient register. After showing that the census and National Health Service migration flows can be reasonably combined, elderly migration flows between groupings of local authority districts by age, sex and health status for the 2000–2001 and 2003–2004 periods are estimated and then analysed to show how the patterns have changed. By combining registration data with census data, we can provide recent estimates of detailed elderly migration flows, which can be used for improvements in social planning or policy.     

Acquired immune deficiency syndrome projections in England and Wales: interplay of methodology and data

Since the late 1980s, regular monitoring of the human immunodeficiency virus epidemic in England and Wales has been carried out through the work of successive national working groups. One of their tasks has been to provide short-term projections of the incidence of acquired immune deficiency syndrome. In this paper the data and methods used in this projection work are reviewed and results critically assessed with the aim of highlighting the strong interaction between methodological developments and data acquisition.     

Acquired immune deficiency syndrome predictions for England and Wales (1992-97): sensitivity analysis,information, decision

Some implications of the use of the back-calculation method for estimating future trends in HIV infections and AIDS incidence in England and Wales are explored. "This paper explores in...detail some aspects of the latest projections which have only been hinted at in the report published by the Public Health Laboratory Service in 1993. The value of additional information on the HIV epidemic in discriminating between different, otherwise equally plausible, scenarios is demonstrated. The role of the backcalculation approach in determining whether, and how, the incubation distribution has been affected by increased uptake of pre-AIDS prophylaxis and treatment is discussed."     

Estimation of infant mortality rates categorized by social class for an Australian population

"This paper reports a method of deriving simultaneous confidence intervals for [Australian] infant mortality rates based on a birth sample rather than the birth population. The large sample size employed enables the use of asymptotic multivariate techniques....[The authors find that] where the population distribution of a characteristic such as social class is not known, confidence intervals can be estimated for rates based on the distribution of this characteristic in a sample of that population."     

Premature mortality in Poland by region

Using an indicator of premature mortality developed by B. Ts. Urlanis, the author calculates mortality differentials for Poland by administrative region (voivodship).