Probabilistic mortality forecasting with varying agespecific survival improvements
 Christina BohkEwald^{1}Email author and
 Roland Rau^{1, 2}
DOI: 10.1186/s4111801600178
© The Author(s) 2016
Received: 22 July 2016
Accepted: 22 July 2016
Published: 12 January 2017
Abstract
Many mortality forecasting approaches extrapolate past trends. Their predictions of the future development can be quite precise as long as turning points and/or ageshifts of mortality decline are not present. To account even for such mortality dynamics, we propose a model that combines recently developed ideas in a single framework. It (1) uses rates of mortality improvement to model the aging of mortality decline, and it (2) optionally combines the mortality trends of multiple countries to catch anticipated turning points. We use simulationbased Bayesian inference to estimate and run this model that also provides prediction intervals to quantify forecast uncertainty. Validating mortality forecasts for British and Danish women from 1991 to 2011 suggest that our model can forecast regular and irregular mortality developments and that it can perform at least as well as other widely accepted approaches like, for instance, the LeeCarter model or the UN Bayesian approach. Moreover, prospective mortality forecasts from 2012 to 2050 suggest gradual increases for British and Danish life expectancy at birth.
Keywords
Mortality forecasts Rates of mortality improvement Prediction intervals Bayesian inference Coherent mortality forecastsIntroduction
Few activities of demographers receive as much attention from society as population forecasts. Many areas of public policy are affected by the increasing number of the elderly in a population as a result of improved chances of survival (Preston and Stokes 2012). Financing oldage pensions and health care and the provision of longterm care are only the tip of the iceberg. However, not only public policy is affected. Private companies also require reliable estimates for future mortality. Pension funds represent the most obvious example (OECD 2011; Soneji and King 2012).
Despite their great societal relevance, the recent track record of demographic forecasts is not praiseworthy. Even though longer time series became available as well as presumably better methodology and more computing power were onhand, Keilman (2008) concluded for European countries that “Demographic Forecasts Have Not Become More Accurate Over the Past 25 Years.” Projecting only mortality—as one of the three parameters, besides fertility and migration, that determine the size of a population and shape its agestructure—does not yield satisfactory results either (Keilman 2008, p. 146).
The seminal method of Lee and Carter (1992) generates robust mortality forecasts using a parsimonious principal component model that captures age and period effects. Numerous extensions of this model have been developed to increase the predictive performance (see for an overview, e.g., Booth (2006), Booth et al. (2006), Booth and Tickle (2008), Shang et al. (2011, 2012), and Butt and Haberman (2010a). There are also more flexible approaches such as the Pspline approach of Currie et al. (2004), approaches using gaps from bestpractice life expectancy at birth (e.g., Torri and Vaupel 2012), and approaches using cause of death data (e.g., Oeppen 2008), as well as approaches that use information from exogenous factors such as tobacco smoking (e.g., Janssen et al. 2013), GDP, and alcohol consumption (e.g., Girosi and King 2008; French and O’Hare 2014). Latest developments in mortality forecasting increasingly include Bayesian inference (see information below), and they account for the distribution of the age at death in order to capture detailed patterns of mortality (Janssen and de Beer 2016; Basellini and Camarda 2016).
Although many of these approaches can yield better forecasts than the original LeeCarter model, it is still challenging to forecast unsteady mortality developments such as turning points in the longterm trend (Coelho and Nunes 2011) or the aging of mortality decline (Horiuchi and Wilmoth 1995). Nevertheless, some recently developed approaches address this issue using different modeling strategies. For example, they extend the LeeCarter predictor structure to let the age pattern rotate with time (e.g., Li et al. 2013; Ševčíková et al. 2016), use multiple principal components in the productratio functional approach (e.g., Hyndman et al. 2013; Hyndman and Ullah 2007), use mortality trends of multiple populations (e.g., Russolillo et al. 2011; Li and Lee 2005), or use rates of improvement rather than death rates to forecast variable mortality changes (e.g., Haberman and Renshaw 2012; Mitchell et al. 2013).
The objective of this paper is to propose a model that newly combines three of these different modeling strategies in order to forecast mortality for developed countries, particularly in the presence of changing longterm trends and dynamic ageshifts of survival improvement. The modeling strategies include the following:
(a) Rates of mortality improvement (ROMIs)
As noted above, Mitchell et al. (2013) and Haberman and Renshaw (2012) use ROMIs to forecast variable mortality changes. They take the predictor structure of the LeeCarter framework and replace the log death rates with their corresponding rates of improvement. Both approaches argue that they can yield better forecasting results than the original LeeCarter model and some of its variants. We also use ROMIs in the proposed model; we define them as the timederivative of agespecific death rates. In our opinion, there are two main reasons that justify using ROMIs. Firstly, while the actual level of death rates determines current values of life expectancy at birth, it is the agespecific rate of change that determines the development into the future. It has been demonstrated in the past that the pace of survival improvement is rather independent of the current level of mortality (Vaupel 1997; Kannisto et al. 1994). Secondly, it is now well established that life expectancy at birth is rising for more than 170 years (e.g., Oeppen and Vaupel 2002; White 2002; Tuljapurkar et al. 2000; Vallin and Meslé 2009). Despite the linear pattern in the increase of life expectancy at birth, mortality did not decrease at all ages simultaneously. In many (highly) developed countries, infant and childhood ages contributed most to the increase in life expectancy at birth in the nineteenth and the early twentieth century; nowadays, life expectancy at birth rises primarily because of reductions in mortality among the oldestold (Christensen et al. 2009). Death rates are falling now at very advanced ages (e.g., above 85 and even above 90), where mortality was often considered to be fixed (Rau et al. 2008; Vaupel 2010), at such a rapid pace that even purely datadriven models were unable to capture the trend (e.g., Currie et al. 2004). We expect that modeling ROMIs instead of death rates can enable the proposed model to capture underlying mortality dynamics (often better than benchmark models); incorporating the aging of mortality decline might help to increase the quality of predictions.
(b) Combine mortality trends of multiple countries
Recent studies give evidence that it can be advantageous to jointly forecast mortality of multiple populations. For instance, Cairns et al. (2011) forecast comoving mortality for two populations, whereas Russolillo et al. (2011) and Li and Lee (2005) forecast mortality of a single country using a shared trend among a group of countries. The assumption behind this methodology is convergence of mortality. It can be justified for a group of countries experiencing sufficiently homogeneous mortality such as (highly) developed countries or western industrialized countries. A reason for converging mortality could be, for instance, that populations who are in the same stage of health transition belong to one group. The theory of Health Transition has been proposed by Frenk et al. (1991), and it is used, for example, by Vallin and Meslé (2004) in order to explain converging and diverging mortality trends for multiple countries across time. The proposed model can, in this regard, use the mean mortality trend of reference countries (RCs) to complement the mortality trend of a single country of interest (COI). This is especially useful if a forecaster anticipates that mortality in a COI will increasingly converge to mortality in RCs (that belong to the same group). For instance, the longterm increase of Danish female life expectancy at birth has been interrupted by a period of almost stagnation during the 1980s and early 1990s. A simple extrapolation of this stagnating trend would lead to an underestimation of the progress in mortality that has been observed thereafter. The proposed model could capture this turning point, if it modeled the Danish mortality trend to increasingly converge to the one of Sweden. Sweden also belongs to western countries, but it has experienced lower mortality and greater survival improvements than Denmark at the same time. Supporting this argument, Vallin and Meslé (2004) suggested that Denmark may enter a new phase of convergence to levels and trends of mortality in western industrialized countries (such as Sweden) from the early 1990s onwards. Hence, in the presence of (anticipated) longterm trend changes, we think that it can be worthwhile for a model to (be able to potentially) assume that the trend of a COI will increasingly approach the trend of RCs (that belong to the same group as the COI) during forecast years. The anticipation of longterm trend changes, and the selection of RCs is, incontestably, subjective. How subjective information and expert judgment can be differentiated in (demographic) forecasting is discussed, for example, in general in Armstrong (2001) and more specifically in Billari et al. (2014a). In order to avoid an arbitrary character of choosing RCs, a possible strategy may be to take a larger group of populations that share major mortality risk factors and survival conditions such as western industrialized countries. This strategy would be similar to, for example, the applied world mean in the Bayesian approach of Raftery et al. (2013) or to the mean of leading countries that is used in the mortality forecasts of EUROSTAT, EUROPOP 2010 and 2013 (European Commission 2014).
(c) Probability statements
We use simulationbased Bayesian inference to estimate and run the proposed model; it automatically captures and quantifies forecast uncertainty with probability statements. Despite the early contribution of Törnqvist in the middle of the twentieth century (Alho and Spencer 2005), generating mortality forecasts with probability statements using (simulationbased) Bayesian inference is still relatively new. For instance, Girosi and King (2008) and King and Soneji (2011) use Bayesian inference to forecast agespecific mortality (by cause of death) as a dependent variable that is influenced by multiple explanatory variables like sex, location, and GDP. Raftery et al. (2013) revise the traditional deterministic approach from the UN to forecast life expectancy at birth. This model has recently been revised by Ševčíková et al. (2016) in order to account for aging of mortality decline. The UN approach combines countryspecific and overall country information with a time series approach. Czado et al. (2005), Pedroza (2006), and Kogure and Kurachi (2010) use Bayesian inference to revise the LeeCarter model in order to address problems with, for instance, erratic data, projection uncertainty, and missing data. Wiśniowski et al. (2015) propose a comprehensive Bayesian population projection model that also uses revised versions of the LeeCarter model to forecast vital rates and migration. Billari et al. (2012; 2014b) propose a Bayesian model to forecast a population probabilistically, using opinions from experts regarding the future development of vital rates as input data in order to determine, e.g., their (expected) median values, marginal variability, and correlation structure across time.
Using the three modeling strategies, i.e., ROMIs, converging mortality, and Bayesian inference, the proposed model is supposed to tackle two issues (that are particularly important in developed countries): Firstly, dynamic age shifts in mortality decline and, secondly, turning points in longterm trends in forecast years.
The remainder of our paper is organized as follows: First, we formally introduce the mortality forecasting model, i.e., its input, core, and output. Thereafter, we apply the proposed model to forecast mortality for women and men in the UK and Denmark. We select these two countries, because they represent two general forecasting situations in terms of the presence of turning points in the longterm mortality trend. While the increase in life expectancy at birth was rather uniform for British women and men, it included longterm trend changes for Danish women and men. We use these different mortality trends to assess the performance of the proposed model in validating forecasts, and to compare its predictive ability with that of the model of Lee and Carter (1992), some of its variants (Renshaw and Haberman 2003, 2006; Li and Lee 2005), the Pspline approach (Currie et al. 2004), the UN Bayesian approach (Raftery et al. 2013), and the Eurostat EUROPOP2010 forecast (European Commission 2011). Finally, we summarize and discuss our main findings in the last section.
Methodology of our model
The objective of the proposed model is to forecast mortality for developed populations, even in the presence of changing longterm trends in mortality and dynamic ageshifts of survival improvement. Moreover, the model should be adjustable to different forecast conditions such as a rapid or slow convergence of mortality between a COI and RCs. This flexibility is achieved using optional features (or tools) that can be applied solely or in combination.
Model input
Core models
We have two optional core models (linear and exponential), which we can use to forecast survival improvements by age and calendar year. Since we use Bayesian inference to run our core models, they automatically model coherence of mortality change among adjacent ages. This ensures that mortality of neighboring ages declines at a similar pace and that mortality gradually increases at adult ages each year. Another advantage is that they capture and quantify forecasting uncertainty, i.e., they give information about the spread and likelihood of our outcome.
Core model 1 (linear model)
These prior specifications are not fixed; they can be adjusted to specific forecast situations. For example, if a forecaster expected only positive survival improvements, she or he could set the uniformly distributed priors for the overall means of the intercepts and slopes to a range of nonnegative values (for more information please see additional notes in the “Model and prior specification” section).
Core model 2 (exponential model)
Implementation
To forecast ROMIs with our core models, we use simulationbased Bayesian inference, i.e., we use the Gibbs sample algorithm (Geman and Geman 1984) to explore the forecasting distribution simulatively. A stationary distribution is reached, if different runs of the Markov chain Monte Carlo (MCMC) simulation converge reliably to similar posterior estimates (Koller and Friedman 2009, p. 509).
To compute and analyze the forecasting distribution, we use the statistical software R (2012) and Just Another Gibbs Sampler (JAGS), a freely available program that can be deployed for Bayesian analysis (Plummer 2012). We use the Rpackages rjags (Plummer 2015) and R2jags (Su and Yajima, 2015) to interface between R and JAGS.
JAGS requires certain input data to compute the forecasting distribution of the ROMIs using our core models. Hence, we define our core models in a specific programming language for JAGS. We also provide basic data in R that will be used from JAGS to execute the models. To this basic data belong, for instance, the observed ROMIs, the number of single ages, and the number of forecast years, as well as the agespecific coefficients θ _{1,x } and θ _{2,x } for the exponential model. Next to this basic data, we determine several execution properties like the random number generator, the number of parallel chains, the number of iterations, and the number of thinning, as well as the length of the burnin period. In general, it can be helpful to fit the forecasting models with a large number of chains and iterations to explore the parameter space exhaustively. We use the Rpackage R2jags to execute parallel chains with JAGS; this can speed up the execution of many iterations. Thinning, by means of saving only each jth iteration, can be used to avoid dependencies between adjacent iterations (autocorrelation). To analyze the outcome, we only use the iterations of a chain that has converged to the forecasting distribution. This also means that we use the burnin period to identify the parameter values of iterations before a chain converges to a constant mean and to exclude them from outcome analysis (King 2012). Running the Gibbs sampling algorithm, as, for instance, described in the “The proposed model” section, we obtain all model parameter estimates simultaneously, and we forecast the mortality trends and their uncertainty coherently. Hence, both core models provide probabilistic mortality forecasts.
Model output
These forecasted death rates can then be used for further mortality analysis, such as life table calculations. To summarize mortality in a population, we use, for instance, the forecasted death rates to compute life expectancy at birth.
Additional notes
Assuming mortality convergence to capture longterm trend changes
As already motivated in the “Introduction” section, converging mortality between a COI and RCs is a modeling strategy (of the proposed model) that can be optionally applied to forecast longterm trend changes for a COI. In terms of life expectancy at birth, convergence could either mean an accelerated increase for the COI to approach larger values of RCs or a slowdown for the COI to approach smaller values of the RCs. However, the anticipation of trend changes, the selection of RCs, and the combination (or weighting) of multiple population trends are, incontestably, subjective.
Converging mortality implies that a COI and RCs share similar mortality risk factors and survival conditions. Broadly defined groups of countries with similar characteristics may be (highly) developed countries and western countries (Russolillo et al. 2011; Li and Lee 2005; European Commission 2014). Moreover, the Epidemiological Transition by Omran (1971) and the Health Transition by Frenk et al. (1991) may serve as a theoretical basis to justify the assumption of mortality convergence between a COI and RCs. If a COI and RCs belong to the same group and the COI simply lags some years behind and is expected to catch up in coming years, we recommend to use mortality trends of vanguard RCs (of the same group) in order to close the gap and to forecast converging mortality between the COI and RCs. This also works in the opposite direction: If the COI is some years ahead and it is expected to slowdown in coming years, we recommend to use mortality trends of lagging RCs (of the same group) in order to forecast converging mortality.
Mortality trends of the COI and RCs can be combined outside of the core models in order to determine different pathways of mortality convergence; a simple way to do this is to gradually increase the weight of the RCs to one and to gradually decrease the weight of the COI to zero in the forecast years. More sophisticated weighting schemes could be designed, including time delays for variable onsets of converging mortality in the years to forecast.
Model and prior specification
Since the model specifications in the “Core models” section aim to be convenient to forecast mortality for many (developed) populations, we use weakly informative prior distributions to “regularize” the posterior distribution and to keep it roughly within reasonable bounds (Gelman et al. 2014, p. 52). Although we are aware that choosing weakly informative priors is fraught with difficulties, the experiences we have made with uniform distributions have been quite positive. Moreover, prior sensitivity analyses in the “Diagnostics” section indicate that the results (or forecasts) of our model are only marginally sensitive to various prior specifications, including other distributions, such as the Beta, Gamma, and Normal distribution. However, users of our model could adjust the priors so that they better reflect the given mortality conditions and their expectations for a population of interest.
Potential limitations and future enhancements
The modeling of dynamic mortality changes via survival improvements and mortality convergence via RCs makes our model particularly suitable to forecast mortality (1) extrapolatively, if mortality in a COI is expected to continue regularly, or (2) extrapolatively and explanatorily, if mortality in a COI is expected to change and to approach the mortality of RCs. This strength also exposes the potential weakness of our model, since it is less suitable to forecast mortality for a country, if a past trend is expected to change in a way that has not yet been observed in any population. This might be the case in, for instance, East Germany and Russia, two countries which experienced an unprecedented fall/rise in mortality after the dissolution of the Soviet Union. To overcome this limitation, we could use expected or target mortality change schedules, a strategy similar to (optimal) model life tables, which have been suggested by the United Nations (1982) for developing countries. Another possible enhancement would be a datadriven choice of (1) RCs and (2) combinations of multiple mortality trends. The model could find a compromise between trends of the COI and RCs via, for example, time series cointegration. Similarly to King and Soneji (2011), we could also add a covariate representing mortality improvement in a RC with a lag of n years (instead of a risk factor such as obesity or smoking 25 years back in time). Despite the fact that such model specifications would reduce the subjective character of our proposed model, they would probably impede the modeling of expected longterm trend changes in forecast years if past data do not already incorporate them. This is mainly because such specifications would identify similar (or common) trends rather than expected trend changes. Future analyses might reveal the pros and cons of such alternative model specifications.
Application of our model
In this section, we apply the proposed model to forecast mortality of British and Danish women and men, and we compare its predictive ability with those of other, wellestablished models. We are aware of the fact that such comparisons are insufficient to fully assess the accuracy of the applied approaches (Hyndman et al. 2013); however, they are useful to give a first impression how each approach deals with regular mortality developments in the UK and irregular mortality trends in Denmark.
An obvious choice for comparative purposes is the model of Lee and Carter (1992) due to its worldwide acceptance and application in the field of mortality forecasting (Booth et al. 2006; Shang et al. 2011). Numerous enhancements of this model have been proposed since its introduction about 20 years ago. For our comparison, we select three generalizations suggested by Renshaw and Haberman (2003; 2006) who add terms regarding cohort and extraperiod effects. We also employ the coherent variant of Li and Lee (2005). Apart from the class of those LeeCarter models, we also compare our model with the Bayesian approach of Raftery et al. (2013) that is used by the UN Population Division for the World Population Prospects 2012 (United Nations 2013) and with the one proposed by Currie et al. (2004) that is based on a twodimensional, nonparametric smoothing using Psplines (Eilers and Marx 1996).
All models are applied and tested using the different mortality trends in the UK and Denmark. British women and men feature a rather regular mortality development with a stable increase in life expectancy at birth in the last decades. Danish women, by contrast, experienced a period of virtual stagnation during the 1980s until the middle of the 1990s, caused by the widespread prevalence of smoking of women born between the two World Wars (LindahlJacobsen et al. 2016; Jacobsen et al. 2002; Christensen et al. 2010). In addition, the slow increase in life expectancy at birth for Danish men since the 1960s has strengthened in the early 1990s. The detection of such structural changes in mortality trends has been analyzed recently (Vallin and Meslé 2009; van Berkum et al. 2013) and poses additional obstacles to forecast mortality accurately.
We generate validating and prospective mortality forecasts for British and Danish women and men with all models. Since the base period has a large effect on mortality forecasts (Stoeldraijer et al. 2013; Janssen and Kunst 2007), we use at least as many reference years as we have years to forecast. Moreover, we take the same base year in the validating and prospective forecasts in order to include similar past trends. The validating forecasts from 1991 to 2011 are based on data from 1965 to 1990. They can be used to assess the predictive ability of each model by simply comparing the estimates of the models with the observed values. We expect that the models perform similarly well in the British case due to the continuation of the past trend in the forecast years. By contrast, we expect greater performance differences between the models in the Danish case due to the longterm trend change in the forecast years. This accelerated increase in Danish female and male life expectancy at birth might be better captured by the coherent approaches (i.e., the model of Li and Lee (2005) and the proposed model) that can include faster mortality reductions of other populations (that still belong to the same group such as highly developed countries). The prospective forecasts from 2012 to 2050 are based on death rates between 1965 and 2011. As we do not know how mortality will develop in the future, we cannot compare forecasted and observed mortality data. Alternatively, we compare the mortality forecasts of our model with those of other approaches and agencies like Eurostat.
Comparing the performance of such different methods, which use extrapolation, explanation, and/or expectation to forecast mortality, can be fair as long as concrete assumptions like the base period and the length of the forecast horizon are equal (Stoeldraijer et al. 2013). Since the concrete assumptions are equal in the British and Danish mortality forecasts, the approaches only differ in their method and in their methodrelated assumptions, which can be, for instance, the transformation of death rates regarding age groups (of single or 5 years of age), smoothing, differentiating, the usage of fitted or observed jumpoff rates, or the assumption of converging mortality. Since all these approaches forecast (total) mortality, use the same death rates from the Human Mortality Database (2013) for the same base period and forecast horizon, we reckon that differences in the forecasts are only due to these methodrelated differences. In the past, such comparisons have been conducted only for extrapolative methods (Booth et al. 2006; Shang et al. 2011) but also for extrapolative and explanatory methods (Stoeldraijer et al. 2013), including methods that even use etiological data such as smoking and nonsmokingrelated mortality (Janssen et al. 2013; Wang and Preston 2009).
Data and parameter settings
The proposed model
To forecast mortality with the proposed model for women and men in the UK and Denmark, we take death counts and exposure times for single ages (0 to 100+) and years from the Human Mortality Database (2013). Following Vallin and Meslé (2004), we assume that mortality of Danish women and men will increasingly converge to the lower overall mortality of other highly developed countries. In the validating forecast, we take mortality data from Sweden, and in the prospective forecast, we also include data from France, Italy, and Japan. We compute ROMIs for each agespecific death rate over time; the death rates are smoothed using Psplines (Currie et al. 2004; Eilers and Marx 1996). We then forecast the survival improvements with our second core model via JAGS by letting the Gibbs sample algorithm run with five parallel chains for a total of 5200 iterations after a burnin period of 200 iterations. The runlength and related settings rely on the RafteryLewis diagnostic and trace plots, which are presented in the “Diagnostics” section. To avoid dependence between adjacent trials, we only save each fifth iterate for inferences. In addition, we exclude implausibly low and high forecasts for ROMIs by setting thresholds, i.e., they can neither fall below 0.005 nor can they exceed 0.035. These thresholds rely on observed values (see Fig. 1), and they imply a mortality decline for all ages at a minimum of 0.5% and at a maximum of 3.5%. An annual mortality improvement of 3.5% is extraordinarily high and means that mortality can be reduced by half in less than 20 years. In case the model approaches minimum or maximum ROMIs, the forecasted death rates still decline with time. Appendix Table 3 summarizes all parameter settings for our mortality forecasts.
Model of Lee and Carter (LC)
The mortality forecasts of the LC model are estimated with the Rpackage demography of Hyndman et al. (2015), which requires death rates and exposures by age and time, arranged in a demogdata object, as input. In the lca function, we set the parameter adjust to dt to adjust the time index k _{ t } to the total number of deaths when the model is refitted. We then use the forecast function to forecast mortality with the estimated model and to compute prediction intervals.
Models of Renshaw and Haberman (RH)
We generate mortality forecasts with three extensions of the LC model proposed by Renshaw and Haberman (2003; 2006), applying the Rpackages ilc (Butt and Haberman 2010b) and demography (Hyndman et al. 2015). We fit the three models, labeled as h0, h1, and h2, with the lca.rh routine that needs death rates and exposures, arranged in a demogdata object, as input. For most parameters, we use the default settings. Exceptions are parameters determining the type of the model and the error structure; we set them to h0, h1, or h2, and to Gaussian or Poisson. In addition, we set the sparsparameter to the recommended value of 0.6, which is intended to smooth the input data. To forecast the fitted models, we use the forecastfunction, setting the jump.choiceparameter to actual, which produces more plausible mortality forecasts than setting it to fit. The output contains forecasted death rates and life expectancies at birth. In our comparative analysis, we only take the parameter settings for each of these three models, which achieve the best forecasting results.
Pspline approach of Currie et al. (Pspline)
The Pspline approach of Currie et al. (2004) is used to smooth and forecast mortality over age and time; it models death counts with the natural logarithm of the exposed population as an offset in a Poisson setting. We employ the software package MortalitySmooth by Camarda (2012). Two parameters are of crucial importance: the smoothing parameter(s) λ and the order of the penalty. The λs are found by optimizing the BIC on a large grid of potential λ values. Specifying the correct order of the penalty is less straightforward, as pointed out in the paper (Currie et al. 2004, p. 289, 297). An order of the penalty of 1 refers to future mortality at a constant level, 2 to improvements at a constant rate, and 3 to an accelerating rate. After experimentation, we choose the default order of the penalty of 2, which gives the best results in our validating framework.
Coherent model of Li and Lee (LCcoherent)
To estimate the coherent model of Li and Lee (2005), we use the webbased platform LCFIT that is hosted on the website of Berkeley’s demography department. The interface requires mortality rates and population counts for at least two countries. We combine the Danish data with death rates and exposures from Sweden in the validating forecast, and, additionally, with mortality data from France, Italy, and Japan in the prospective forecast. Moreover, experimentation has shown that the default settings of the optional parameters produce the best results.
Bayesian approach of Raftery et al. (BayesLife)
We use the Rpackage bayesLife (Ševčíková and Raftery 2015) to forecast mortality with the Bayesian approach of Raftery et al. (2013). Applying the run.e0.mcmc and the e0.predict routine with default parameter values, we simulate 160,000 iterations using three chains. The exploration of the trajectories of future life expectancy at birth yields quantiles for quinquennial data, i.e., in the validating forecasts, this approach provides life expectancy at birth for the 5year periods 1990–1995, 1995–2000, 2000–2005, and 2005–2010 of which we take the midpoints 1993, 1998, 2003, and 2008.
Validating forecast
Prediction intervals
Empirical frequencies to assess calibration of prediction intervals in the validating mortality forecasts for British and Danish women
Prediction interval:  United Kingdom  Denmark  

95%  80%  95%  80%  
Our model  100  100  91  81 
LCHynd  100  100  100  86 
Pspline  100  91  62  33 
LCcoherent  –  –  100  95 
RHh0  5  0  0  0 
RHh1  100  95  100  47 
RHh2  100  24  5  0 
Empirical frequencies to assess calibration of prediction intervals in the validating mortality forecasts for British and Danish men
Prediction interval:  United Kingdom  Denmark  

95%  80%  95%  80%  
Our model  100  100  95  95 
LCHynd  76  62  14  0 
Pspline  100  100  71  67 
LCcoherent  –  –  29  24 
RHh0  20  0  0  0 
RHh1  81  29  48  5 
RHh2  0  0  0  0 
Median forecasts
In Fig. 3, the forecast errors are displayed for each model below the two panels. In the case of the UK (left panel), the Pspline model and our model provide a satisfactory fit; their errors fluctuate around zero, resulting in mean absolute errors of 0.17 and 0.16 years, respectively. Considering that (record) life expectancy at birth increases annually at a pace of about 0.25 years, this deviation is negligibly small. By contrast, the errors of the LC model and its extensions of RH are somewhat greater and they accumulate with time. Their mean absolute error ranges between 0.43 and 0.57 years, and they underestimate life expectancy at birth in 2011, the end of our validating forecasting period, by 1.30 to 1.58 years.
As pointed out previously, Denmark’s life expectancy at birth (right panel) is challenging to forecast due to a trend change in the forecast years. Surprisingly, the nonparametric Pspline approach produced the largest forecast errors. Neither modifying the order of the penalty nor adding weights, which change over time, improves the Pspline forecast. The LC model and its variants of RH perform better; the approaches assuming converging mortality capture the trend change in the forecast years even better. That is why the maximum absolute errors of the coherent forecast of Li and Lee (2005), 1.02 years, and of the proposed model, 0.81 years, are so low compared to the other approaches. The results are also given for British and Danish men in Figure 9 in the Appendix; the pattern of the errors are similar to the results for women; however, they are more pronounced for men. In addition, Figure 10 in the Appendix depicts the coherent forecasts for Danish women and men assuming a convergence to mortality of Sweden, France, Italy, and Japan (as in the prospective forecasts in the next section). These forecasts show that both the coherent LC model and the proposed model are sensitive to the choice of RCs. That the proposed model is sensitive to the choice of RCs is also shown, for example, in a mortality forecast for the USA (Bohk and Rau, 2016). Moreover, the forecasts of remaining life expectancy at age 65 are given for the proposed model in Figure 11 in the Appendix.
Prospective forecast
Prediction intervals
Median forecasts
Comparing the prospective mortality forecasts reveals several aspects. First, all models predict that the increase in life expectancy at birth is likely to continue. Models applying the LC predictor structure forecast stronger improvements for British than for Danish women. For example, they predict that British women will gain on average seven additional years of life until 2050, while Danish women are forecasted to gain only five additional years of life. This is probably due to the weak survival improvements during the 1980s and early 1990s in Denmark. Thus, the trend in the base period has a decisive influence on a forecast, particularly for purely datadriven methods that extrapolate only past trends. The LCcoherent model and the proposed model forecast a stronger growth in Danish life expectancy at birth than the other models; that is because they assume that Danish mortality will converge to the lower mortality trends of Swedish, French, Italian, and Japanese women. Additional years of life until 2050 are forecasted to be 8.6 years according to our model and 7 years according to the LCcoherent model. Although both approaches combine the same mortality trends in their forecasts, the transition between observed and forecasted values appears to be more plausible (or smooth) for our model than for the coherent model; at least we were unable to remove this jumpoff bias using the webbased platform LCFIT. Second, it is striking that our model is not restricted to predict only moderate gains or only large gains in life expectancy at birth. In the British case, our model forecasts moderate gains, and in the Danish case, it forecasts strong increases in life expectancy at birth compared to the other approaches. By contrast, the UN Bayesian approach and the LC model provide rather conservative forecasts for British and Danish women. Third, the forecasted values for life expectancy at birth vary considerably between the models. For example, the Pspline approach forecasts one of the greatest values for life expectancy at birth in 2050: For British women, it is 92.18 years, and for Danish women, it is 88.97 years. By contrast, the LC model forecasts life expectancy at birth to be relatively low in 2050: For British women, it is 87.95 years, and for Danish women, it is 86.36 years. Only the UN Bayesian approach forecasts even smaller values for life expectancy at birth: For British women, it is 86.54 years, and for Danish women, it is 85.12 years (in 2048). Fourth, each model provides prediction intervals to capture and quantify forecast uncertainty. For example, the 95% prediction interval of our model and of the Pspline approach is relatively wide: In 2050, the life expectancy at birth of British women is estimated to range between 85 and 96 years according to our model and between 87 and 96 years according to the Pspline approach. By contrast, the other models provide smaller prediction intervals. For example, the LC model predicts British female life expectancy at birth to range between 85 and 90 years with a probability of 95%. The prospective median forecasts are also given for men in Figure 12 in the Appendix; the forecasted trends are similar to those described for women.
Diagnostics
In this section, we assess if the Gibbs sampler converged to the correct posterior distribution (via trace plots and autocorrelation functions), if the runlength is sufficient (via the RafteryLewis diagnostic (Raftery and Lewis 1992)), as well as if the forecasts are (in) sensitive to the prior specifications.
Appendix Table 4 lists the results of the RafteryLewis diagnostic, i.e., the number of iterations needed for each model parameter to obtain an accurate estimate for the 0.025 quantile with a probability of 0.95. For the UK as well as for Denmark, the estimated runlength N fluctuates between 3561 and 3741 for all model parameters, and the dependence factor I fluctuates between 0.951 and 0.999. As neither the runlength N nor the dependence factor I has exceptionally high or low values for certain model parameters, we conclude that 5000 iterations will be sufficient to estimate an accurate projection outcome with our model.
A simple prior sensitivity analysis suggests that our British and Danish mortality forecasts are relatively insensitive to the (rather weakly informative) prior specifications. To demonstrate this, we repeatedly generate validating forecasts with our model using different prior specifications, i.e., we use the Uniform, the Beta, and the Normal distribution with different parameter values to determine various levels for their location and spread. Appendix Figures 15 and 16 show that these different prior specifications result in (very) similar marginal posterior densities and, beyond that, Appendix Figure 17 shows that these prior specifications also yield (very) similar forecasts for British (left) and Danish (right) female life expectancy at birth from 1991 to 2011. These findings suggest that the results of our model are mainly datadriven.
Discussion
Advances in mortality forecasts will become even more important in the future than they are already today. In many highly developed countries, improving chances of survival are the main driver of population aging (Preston and Stokes 2012); the increase in life expectancy at birth is due to mortality reductions at all ages, though the highest gains in mortality reductions advance to successively older ages (Christensen et al. 2009). Some forecasts underestimate this ongoing progress in mortality reductions from younger to older ages (Ševčíková et al. 2016; Li et al. 2013). In the past, this failure would have meant to underestimate mortality of people at working ages; in the future, it would probably lead to an underestimation of mortality of people aged 65 and older (because survival improvements will increasingly reach those ages). This will have an increasing impact on social welfare systems. Underestimating mortality of people at working ages corresponds to having more people who pay premiums. Underestimating mortality of people in retirement ages, however, will result in having more people who receive benefits for longer periods than previously anticipated.
 (1)
Agespecific ROMIs can capture dynamic ageshifts of mortality change over time. ROMIs were employed in recent studies by Mitchell et al. (2013) and Haberman and Renshaw (2012), who used them to extend the predictor structure of the LC model.
 (2)
Converging mortality between a COI and RCs. This can be useful for populations, which undergo irregular mortality developments and that face longterm trend changes in the forecast years. RCs who belong to the same group as the COI, e.g., western industrialized countries, or that will be in the same state of Health Transition as the COI (Frenk et al. 1991; Vallin and Meslé 2004) can be a good choice. Recent works by, for example, Li and Lee (2005), Cairns et al. (2011), and Russolillo et al. (2011), show that it can be worthwhile to jointly forecast mortality for two or more populations, especially when they exhibit similar characteristics concerning health and mortality.
Applying these tools, the validating forecasts of the proposed model mirror the observed life expectancies at birth of women and men in the UK and Denmark with only small deviations. In the case of the UK, the proposed model benefits from using ROMIs, which allow a timevariant age pattern of mortality change. Only the Pspline approach, proposed by Currie et al. (2004), forecasts British mortality with deviations so small that they are similar to those of the proposed model. In the case of Denmark, the proposed model benefits not only from ROMIs but also from convergence between Danish and Swedish mortality trends. According to Vallin and Meslé (2004), we expect that Denmark will approach the lower overall mortality of Sweden in the forecast years. The coherent model of Li and Lee (2005) forecasts Danish mortality with comparably small deviations as the proposed model; that is because we also combined the Danish with the Swedish trend in this forecast.
We interpret these findings as a first indication that the proposed model can generate valid forecasts for developed countries experiencing (ir)regular mortality trajectories; a thorough validation will be part/object of followup analyses possibly including mortality forecasts for more countries using different base periods, forecast horizons, core models, and RCs. Apart from comparing probable future lifespans among countries, such analyses will primarily expose strengths and weaknesses of this methodology (under certain forecast conditions) as well as user guidelines. However, relying on the presented results here, we can give initial recommendations regarding some selection criteria in order to generate mortality forecasts (for developed countries) with the proposed model. First, our model appears to be applicable for forecast horizons up to a length of 50 years. Second, two points appear to be especially relevant for the selection of a core model. The first point is that, contrary to the linear model, the exponential model cannot forecast mortality increases, since the logarithm of negative ROMIs is not defined. The second point is that the linear model extrapolates temporal trends for survival improvements at each age ad infinitum, whereas the exponential model can also forecast a decelerating mortality decline. This may be important for currently steep increases in survival improvements as well as for longer forecast horizons. Third, the subjective choice of RCs is certainly worth discussing. Although purely datadriven methods are objective, they typically fail to forecast trend changes in coming years if they have not been observed in past data; they rather extrapolate the mean trend of the base period. (This can be seen in the validating mortality forecasts for Denmark of the noncoherent LC models in the “Validating forecast” section). To circumvent this problem, the proposed model can assume that mortality of a COI will converge to the mortality of RCs. Regarding the selection of appropriate RCs, we recommend to choose countries that belong to the same broad group as the COI; such a group can be, for example, western industrialized countries. Moreover, the theory of Health Transition by Frenk et al. (1991) or the Epidemiological Transition by Omran (1971) can be used along with major risk factors affecting morbidity and mortality, such as the smoking epidemic (Lopez et al. 1994; Thun et al. 2012; Stoeldraijer et al. 2014; Janssen et al. 2015), to justify or explain the choice of RCs for a COI. Fifth, the proposed model also allows the estimation of purely hypothetical settings, such as how would mortality in country XY develop if it had the (past) mortality experience of country YZ. Such a conditional projection could reveal, for example, how mortality would develop in a COI if the health care conditions of another country would have applied.
In summary, we propose a framework, which appears to be applicable to derive mortality forecasts for developed populations experiencing stable and unstable mortality developments. Since it is uncertain what will happen in the future, the proposed model provides tools that can be switched on or off to adjust to certain forecast conditions. To these optional features belong, for example, the core models (linear or exponential) and the assumption of mortality convergence. This optional (plugin) setting also simplifies further extensions of the proposed model in terms of the integration of additional functions. Despite promising initial forecasts for the UK and Denmark, as well as for Italy, Spain, West Germany (Bohk and Rau 2014), and the USA (Bohk and Rau 2016), our model will be further validated in a next step.
Abbreviations
 COI:

Country of interest
 JAGS:

Just another Gibbs sampler
 LC:

LeeCarter
 MCMC:

Markov chain Monte Carlo
 RC:

Reference country
 RH:

RenshawHaberman
 ROMI:

Rate of mortality improvement
Declarations
Acknowledgements
We would like to thank Ronald D. Lee, Webb Sprague, and Carl Boe for their support to estimate the coherent LC model and Robert Schoen for the discussion of our model during the Annual Meeting of the Population Association of America in Boston in 2014. We would also like to thank the attendees of the Joint Eurostat/UNECE Work Session on Demographic Projections held in Rome, 29–31 October 2013, for their valuable comments.
Funding
The European Research Council has provided financial support under the European Community’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement no. 263744.
Authors’ contributions
Both authors read and approved the final manuscript.
Authors’ information
CBE (research scientist) and RR (principal investigator) have developed this work in the ERCfunded project ROMI at the University of Rostock in the time from 2011 to 2015.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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