Bayesian Forecasting of Mortality Rates for Small Areas Using Spatiotemporal Models

Estimation and prediction of subnational mortality rates for small areas are essential planning tools for studying health inequalities. Standard methods do not perform well when data are noisy, a typical behavior of subnational datasets. Thus, reliable estimates are difficult to obtain. I present a Bayesian hierarchical model framework for prediction of mortality rates at a small or subnational level. By combining ideas from demography and epidemiology, the classical mortality modeling framework is extended to include an additional spatial component capturing regional heterogeneity. Information is pooled across neighboring regions and smoothed over time and age. To make predictions more robust and address the issue of model selection, a Bayesian version of stacking is considered using leave-future-out validation. I apply this method to forecast mortality rates for 96 regions in Bavaria, Germany, disaggregated by age and sex. Uncertainty surrounding the forecasts is provided in terms of prediction intervals. Using posterior predictive checks, I show that the models capture the essential features and are suitable to forecast the data at hand. On held-out data, my predictions outperform those of standard models lacking a regional component.

Smoothing, Decomposing and Forecasting Mortality Rates

The Lee-Carter (LC) model represents a landmark paper in mortality forecasting. While having been widely accepted and adopted, the model has some limitations that hinder its performance. Some variants of the model have been proposed to deal with these drawbacks individually, none coped with them all at the same time. In this paper, we propose a Three-Component smooth Lee-Carter (3C-sLC) model which overcomes many of the issues simultaneously. It decomposes mortality development into childhood, early-adult and senescent mortality, which are described, individually, by a smooth variant of the LC model. Smoothness is enforced to avoid irregular patterns in projected life tables, and complexity in the forecasting methodology is unaltered with respect to the original LC model. Component-specific schedules are considered in projections, providing additional insights into mortality forecasts. We illustrate the proposed approach to mortality data for ten low-mortality populations. The 3C-sLC captures mortality developments better than a smooth improved version of the LC model, and it displays wider prediction intervals. The proposed approach provides actuaries, demographers, epidemiologists and social scientists in general with a unique and valuable tool to simultaneously smooth, decompose and forecast mortality.

Assessing excess mortality in times of pandemics based on principal component analysis of weekly mortality data-the case of COVID-19

The COVID-19 outbreak has called for renewed attention to the need for sound statistical analyses to monitor mortality patterns and trends over time. Excess mortality has been suggested as the most appropriate indicator to measure the overall burden of the pandemic in terms of mortality. As such, excess mortality has received considerable interest since the outbreak of COVID-19 began. Previous approaches to estimate excess mortality are somewhat limited, as they do not include sufficiently long-term trends, correlations among different demographic and geographic groups, or autocorrelations in the mortality time series. This might lead to biased estimates of excess mortality, as random mortality fluctuations may be misinterpreted as excess mortality. We propose a novel approach that overcomes the named limitations and draws a more realistic picture of excess mortality. Our approach is based on an established forecasting model that is used in demography, namely, the Lee-Carter model. We illustrate our approach by using the weekly age- and sex-specific mortality data for 19 countries and the current COVID-19 pandemic as a case study. Our findings show evidence of considerable excess mortality during 2020 in Europe, which affects different countries, age, and sex groups heterogeneously. Our proposed model can be applied to future pandemics as well as to monitor excess mortality from specific causes of death.

Mortality Forecasting with the Lee-Carter Method: Adjusting for Smoothing and Lifespan Disparity

Reliable mortality forecasts are an essential component of healthcare policies in ageing societies. The Lee-Carter method and its later variants are widely accepted probabilistic approaches to mortality forecasting, due to their simplicity and the straightforward interpretation of the model parameters. This model assumes an invariant age component and linear time component for forecasting. We apply the Lee-Carter method on smoothed mortality rates obtained by LASSO-type regularization and hence adjust the time component with the observed lifespan disparity. Smoothing with LASSO produces less error during the fitting period than do spline-based smoothing techniques. As a more informative indicator of longevity, matching with lifespan disparity makes the time component more reflective of mortality improvements. The forecasts produced by the new method were more accurate during out-of-sample evaluation and provided optimistic forecasts for many low-mortality countries.

Mortality forecasting in Colombia from abridged life tables by sex

BACKGROUND

An adequate forecasting model of mortality that allows an analysis of different population changes is a topic of interest for countries in demographic transition. Phenomena such as the reduction of mortality, ageing, and the increase in life expectancy are extremely useful in the planning of public policies that seek to promote the economic and social development of countries. To our knowledge, this paper is one of the first to evaluate the performance of mortality forecasting models applied to abridged life tables.

OBJECTIVE

Select a mortality model that best describes and forecasts the characteristics of mortality in Colombia when only abridged life tables are available.

DATA AND METHOD

We used Colombian abridged life tables for the period 1973-2005 with data from the Latin American Human Mortality Database. Different mortality models to deal with modeling and forecasting probability of death are presented in this study. For the comparison of mortality models, two criteria were analyzed: graphical residuals analysis and the hold-out method to evaluate the predictive performance of the models, applying different goodness of fit measures.

RESULTS

Only three models did not have convergence problems: Lee-Carter (LC), Lee-Carter with two terms (LC2), and Age-Period-Cohort (APC) models. All models fit better for women, the improvement of LC2 on LC is mostly for central ages for men, and the APC model's fit is worse than the other two. The analysis of the standardized deviance residuals allows us to deduce that the models that reasonably fit the Colombian mortality data are LC and LC2. The major residuals correspond to children's ages and later ages for both sexes.

CONCLUSION

The LC and LC2 models present better goodness of fit, identifying the principal characteristics of mortality for Colombia.Mortality forecasting from abridged life tables by sex has clear added value for studying differences between developing countries and convergence/divergence of demographic changes.

Comparing strategies for matching mortality forecasts to the most recently observed data: exploring the trade-off between accuracy and robustness

Background

Given the increased link between retirement age and payments to the development in life expectancy, a precise and regular forecast of life expectancy is of utmost importance. The choice of the jump-off rates, i.e. the rates in the last year of the fitting period, is essential for matching mortality forecasts to the most recently observed data. A general approach to the choice of the jump-off rates is currently lacking.

Objective

We evaluate six different options for the jump-off rates and examine their effects on the robustness and accuracy of the mortality forecast.

Data and methods

Death and exposure numbers by age for eight European countries over the years 1960–2014 were obtained from the Human Mortality Database. We examined the use of model values as jump-off rates versus observed values in the last year or averaged over the last couple of years. The future life expectancy at age 65 is calculated for different fitting periods and jump-off rates using the Lee-Carter model and examined on accuracy (mean absolute forecast error) and robustness (standard deviation of the change in projected e65).

Results

The choice for the jump-off rates clearly influences the accuracy and robustness of the mortality forecast, albeit in different ways. For most countries using the last observed values as jump-off rates resulted in the most accurate method, which relates to the relatively high estimation error of the model in recent years. The most robust method is obtained by using an average of observed years as jump-off rates. The more years that are averaged, the better the robustness, but accuracy decreases with more years averaged.

Conclusion

Carefully considering the best choice for the jump-off rates is essential when forecasting mortality. The best strategy for matching mortality forecasts to the most recently observed data depends on the goal of the forecast, the country-specific past mortality trends observed, and the model fit.

Modeling and Forecasting Mortality With Economic Growth: A Multipopulation Approach

Research on mortality modeling of multiple populations focuses mainly on extrapolating past mortality trends and summarizing these trends by one or more common latent factors. This article proposes a multipopulation stochastic mortality model that uses the explanatory power of economic growth. In particular, we extend the Li and Lee model (Li and Lee 2005) by including economic growth, represented by the real gross domestic product (GDP) per capita, to capture the common mortality trend for a group of populations with similar socioeconomic conditions. We find that our proposed model provides a better in-sample fit and an out-of-sample forecast performance. Moreover, it generates lower (higher) forecasted period life expectancy for countries with high (low) GDP per capita than the Li and Lee model.

Optimizing the Lee-Carter Approach in the Presence of Structural Changes in Time and Age Patterns of Mortality Improvements

Researchers using the Lee-Carter approach have often assumed that the time-varying index evolves linearly and that the parameters describing the age pattern of mortality decline are time-invariant. However, as several empirical studies suggest, the two assumptions do not seem to hold when the calibration window begins too early. This problem gives rise to the question of identifying the longest calibration window for which the two assumptions hold true. To address this question, we contribute a likelihood ratio-based sequential test to jointly test whether the two assumptions are satisfied. Consistent with the mortality structural changes observed in previous studies, our testing procedure indicates that the starting points of the optimal calibration windows for most populations fall between 1960 and 1990. Using an out-of-sample analysis, we demonstrate that in most cases, models that are estimated to the optimized calibration windows result in more accurate forecasts than models that are fitted to all available data or data beyond 1950. We further apply the proposed testing procedure to data over different age ranges. We find that the optimal calibration windows for age group 0-49 are generally shorter than those for age group 50-89, indicating that mortality at younger ages might have undergone (another) structural change in recent years.