A Space Time Extension of the Lee Carter Model in a Hierarchical Bayesian Framework: Modelling and Forecasting Provincial Mortality in Italy

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1 WP October 2013 UNITED NATIONS STATISTICAL COMMISSION and ECONOMIC COMMISSION FOR EUROPE STATISTICAL OFFICE OF THE EUROPEAN UNION (EUROSTAT) Joint Eurostat/UNECE Work Session on Demographic Projections organised in cooperation with Istat (29-31 October 2013, Rome, Italy) Item 14 Multiregional Projections A Space Time Extension of the Lee Carter Model in a Hierarchical Bayesian Framework: Modelling and Forecasting Provincial Mortality in Italy Fedele Greco, University of Bologna Francesco Scalone, University of Bologna

2 Joint Eurostat UNECE ISTAT Work Session on Demographic Projections Rome, October 2013 A Space Time Extension of the Lee Carter Model in a Hierarchical Bayesian Framework: Modelling and Forecasting Provincial Mortality in Italy Fedele Greco Francesco Scalone Department of Statistical Sciences University of Bologna, Italy 1. Introduction The main purpose of this paper is to define a statistical method to model and forecast mortality rates by age and sex for provincial areas in Italy. In this endeavor, we will combine mortality modeling techniques using a Bayesian approach. Standard mortality poorly forecasting performs at a small area level. In fact, when working on single small areas, forecasters have to deal with large variances of direct estimates of the mortality rates, i.e. estimates obtained by exploiting data which refer to each small area separately. We propose a statistical model that allows area level estimates to borrow strength from each other by exploiting spatial association of provincial mortality rates and taking into account temporal correlation. The need for a spatiotemporal model is motivated by an explorative analysis. This approach has become very popular in the disease mapping literature (Kim and Lim 2010, Knorr Held 2000) but, to our knowledge, it has not been employed for modeling and forecasting age sex mortality rates in a demographic framework. In these terms, we redefine an extension of the Lee Carter method as a statistical model accounting for sub populations at provincial geographical level. Adopting a Bayesian approach, Markov Chain Monte Carlo methods (Girosi and King 2008) will be used to fit the model and to sample from the posterior predictive distribution. The outline of this paper is as follows. We first review the most used mortality modeling techniques such as the classical Lee Carter method, its log linear Poisson formulation, the extensions for multiple sub populations and the more recent Bayesian hierarchical models. Then we present the used data providing a brief description of the twentieth century provincial mortality trends in Italy. Afterwards, adopting a Bayesian approach, we define a spatial temporal model, giving details on implementation and prediction. In the final part, some concluding remarks are provided, also presenting possible applications for mortality forecasting.

3 2. Literature review In this section we briefly look at the most recent techniques of modeling forecasting mortality. We don t consider the ones based on pure deterministic and extrapolative methods (see Pitacco et al. 2009, ), focusing on methods that include stochastic components. For the purposes of this paper, we also need to look at methods that forecast mortality in multiple sub group populations. From this point of view, Bayesian hierarchical formulations can be ideal to control for all possible sources of variability (concerning both model parameters and predictions) and to forecast mortality rates on different geographical sub national units, as in the case of the province. The Lee Carter Method Lee and Carter (1992) proposed a method (henceforth the LC method) for modeling long term change in mortality as a function of a unique temporal index. Based on standard time series procedures, the LC method takes into account historical trends and projects distributions of age specific death rates. In order to represent the age specific mortality, Lee and Carter (1992) model the central death rate by using the following formulation. Let denote the central death rate for age x at time t, so we assume the following log linear form:, As Lee and Carter proposed, the s describe the age pattern of mortality averaged over time, whereas the s describe the deviations from the averaged pattern when varies. The change in the level of mortality over time is described by the (univariate) mortality index. The quantity,, denotes the error term, with mean 0 and variance, reflecting particular age specific historical influence that are not captured by the model. However, the LC methodology works as a mere extrapolation based on the fact that the future will be someway like the past, but without taking into account the effects of sudden improvements in survivorship, for example related to new medical discoveries and treatments. In order to predict future age specific mortality rates, Lee and Carter assume that the temporal component can be modeled as a random walk with drift, following an ARIMA(0,1,0) process. As a matter of fact, this stochastic process remains the only source of variability taken into account. Log Linear Poisson Models However, the estimation error of the parameters cannot be taken into account and the proposed model only incorporates uncertainty related to the index forecast (Lee and Carter, 1992). So the prediction intervals may result too narrow. In order to take into account all variability sources, a bootstrap procedure for a Poisson logbilinear formulation of the LC model was implemented (Brouhns et al. 2005). As suggested

4 by Alho (2000), switching from a classical linear model to a generalized linear model is possible. So the original LC method was embedded in a Poisson regression model, which is perfectly suited for age sex specific mortality rates the Poisson distribution arises naturally when data set takes the form of counts (Brouhns et al. 2002). Extensions for multiple sub populations Using the LC method to forecast mortality rates for the two sexes or other sub groups of a population can, however, be difficult. As a matter of fact, it seems improper to forecast mortality for provincial or other sub national populations in isolation from one another. Indeed, closely related populations have similar mortality patterns that do not diverge in the long run. In order to improve the mortality forecasts for individual provinces or other subnational populations, the LC model can be modified by taking into account their membership in a group. While forecasting provincial mortality for Canada, Lee and Nault (1993) propose making use of the same x and k t for each province, only in the case the historical x s do not vary significantly by province. In another application, the LC model is applied to a group of populations, allowing each its own age pattern and level of mortality but imposing shared rates of change by age (Lii and Lee 2005). Bayesian Hierarchical Spatio Temporal Frameworks Turning to the Bayesian approach Czado et al. (2005) proposed a Bayesian formulation of the log linear Poisson to consider all possible sources of variability that affect prediction values and intervals. A further Bayesian formulation of the LC method is given in Pedroza (2006), by using simulation methods. So Markov Chain Monte Carlo (henceforth MCMC) methods are proposed to draw samples from the joint posterior distribution of the parameters and to shape the posterior predictive distribution of the log mortality rates. In particular, the model is fitted and the rates are forecasted making use of the Gibbs sampler (Pedroza 2006). In a Bayesian framework, predicted intervals appear to be more correctly wider than those calculated applying the LC method, since they appropriately incorporate all known sources of variability. As a consequence, forecast variability turns out to be better suited to the observed data. During the last two decades, further hierarchical Bayesian estimates of mortality rates were proposed, based on Binomial or Poisson sampling and taking into account spatial correlation procedure. These hierarchical models are generally based on a Poisson model for the first stage and incorporate covariates by various modeling of the Poisson parameters at the second stage (Kim and Lim 2010). Spatial effects are typically included as random with some distributions where their parameters must be estimated. Maximum likelihood methods are generally used to estimate hyper parameters whereas Bayesian or empirical Bayesian methods are applied to obtain Poisson parameters and rates. In particular, Knorr Held (2000) propose spatio temporal interaction models where the spatial effects are nested within time so that it is possible to take into account how spatial heterogeneity and patterns evolved over time.

5 3. Descriptive analyses First of all, it is worth defining the geo unit of our analysis. In Italy, a province is an administrative division of intermediate level between a municipality and a region. In 2009, on average they count 561,168 persons, whereas the median is equal to 385,729, the first and third quartiles are 232,540 and 586,020, respectively. However, some provinces can be quite different in terms of population dimension. As a matter of fact, ten provinces have more than 1 million inhabitants, whereas 18 have less than 200,000. The two most populated provinces of Rome and Milan count respectively 4,1 and 3,4 million, whereas the least populated ones of Isernia and Ogliastra count 88 and 57 thousands. From this point of view, we want to take into account this variation in population size when we model and predict mortality levels by province. In order to model age specific mortality rates at provincial level, we take into account number of deaths and population amounts by 5 year age groups and provinces in each year from 1993 to We used data collected by the Italian Institute of National Statistics (Istat) referring to 103 Italian provinces in the period between 1993 and The deaths and population series were preliminary reconstructed in order to have time constant provincial borders. Measuring Age Specific Mortality Levels In order to highlight spatio temporal trends of mortality in Italy, we take into account age group specific mortality rates. Let Djtp denote the frequencies of deaths for the t th time period within the j th age group of the p th province. Direct estimates of the age sex specific central provincial death rates are obtained as: The jth age group is a 5 years age class comprising all single one year ages from x to x+4. jtp is the average population in the j th age group at time t in province p: 2 where is the population in the j th age group in December 31 in year t for a given p th province. Thus exposure to risk is therefore given by the population in the j th age group at time t 1 plus the population at time t divided by 2.

6 Mortality Tendencies in Italy at Provincial Level For the sake of brevity, all the figures discussed in this section are referred to the female population. In figure 1, provinces are classified by direct mortality rates level in age group between 0 and 4 for six given years (1992, 1995, 1998, 2001, 2004 and 2007). In each map, data is presented according to a seven intervals classification based on the septiles (7 quantiles). In order to have the same classification in each year, septiles are calculated by taking into account all the observed death rates in each province between 1992 and In these terms, figure 1 represents the general declining trend which occurred in the 0 4 age group during the period in question. Indeed, we can easily see that the first two 1992 and 1995 maps in figure 1 are dominated by the darkest highest mortality categories, since almost all the provinces fell in the highest quantiles. On the other hand, in 2005 and 2007, we can see the clearest lowest mortality categories prevalence. In figure 2, mortality rates in the 0 4 age group are still considered. Nevertheless, category intervals are not constant, since septiles are calculated on death rates ranking of a given single year which the map refers to. In these terms, we can observe the permanence of mortality clustering during the whole period in question, right up to As a matter of fact, higher death rates are concentrated in some visible cluster of provinces, providing evidence of spatial autocorrelation of mortality. In these terms, we demonstrate the importance of spatial structure components when modeling mortality rates for provincial sub populations. It is also important to say that the two tendencies (general mortality decline and clustering persistence) we have just observed for the age group between 0 and 4 are common to the other age classes. To give a further example, in figure 3 and 4, we report direct mortality rates in the age group, creating the interval categories in the same way we did for figure 1 and 2, respectively.

7 Figure 1. Mortality rates in age class 0 4 by province. Colored intervals = Seven quantiles based on the distribution of all mortality rates observed from 1992 to Figure 2. Mortality rates in age class 0 4 by province. Colored intervals = Seven quantiles based on distributions of mortality rates observed in each single given year

8 Figure 3. Mortality rates in age class by province. Colored intervals = Seven quantiles based on the distribution of all mortality rates observed from 1992 to Figure 4. Mortality rates in age class by province. Colored intervals = Seven quantiles based on distributions of mortality rates observed in each single given year

9 Figure 5. APLE Index based on provincial mortality rates by years and 5 age groups for males and females, Males Females Year Year j th Age age class j th Age age class To better demonstrate spatial mortality clustering, figure 5 displays male and female Approximate Profile Likelihood Estimator (APLE) Indexes on direct provincial death rates for different years and age groups. We prefer the APLE index to the more commonly used Moran s I because the former index has shown better behaviors when describing spatial correlation (Li et al., 2007). Stronger spatial autocorrelation is observed for both male and female populations in the first 0 4 age groups on each year from 1992 to The persistence of mortality clustering for infant ages can be due to differences in health care systems and environmental conditions. As a matter of fact, neonatology and pediatric care is organized at district and regional levels and so changing from area to area. Spatial clustering in mortality evidently reduces in younger age classes for both sexes (almost disappearing for males). Direct mortality rates in those ages are very low and thus are more affected by random variations. As a consequence, it is not possible to register spatial autocorrelation. In addition, stronger mortality clustering is evident for males in adult ages (with higher APLE Index levels before 2000), whereas almost no spatial autocorrelation is observed for the oldest ages between 80 and 99. On the contrary, when looking at females, we observe weaker mortality clustering for central adult ages and stronger spatial correlation for oldest age groups. Given the differences in survivorship between the two sexes, higher mortality levels in adult ages significantly and selectively reduce the number of male survivors in oldest age groups. On the one hand, oldest male survivors should be more robust, making biological components prevail on environmental factors and health systems and thus cancelling mortality clustering. In these terms, women in oldest age groups should be less selected and numerous than men and so more affected by geo spatial differences. On the

10 other hand, direct mortality rates for men in oldest age classes are based on very small numbers both at the numerator and denominator, making them extremely sensitive to random fluctuations and thus annulling spatial autocorrelation. 4. Hierarchical Bayesian Models for Space Time Variation In this section, we illustrate a Bayesian hierarchical model in order to take into account the data features described in the previous section. More precisely, the model is intended to capture both temporal and spatial features of the mortality trends, along with spatiotemporal interactions. The model which follows was first proposed by Knorr Held (2000) and has been adopted, with some modification, in several disease mapping applications (see for example: Kim and Lim 2010). As a starting point, we propose separate models for each age class; thus, in order to simplify notation, we drop the subscript j indexing age groups as in previous formulae. Let P tp and D tp denote respectively the exposure to risk and the death counts at year t and province p, t 1,...,T ; p 1,...,P. At the first level of the hierarchy, conditionally on model parameters involved in higher levels, we assume that mortality counts parameter P, i.e.: tp tp Dtp follow independent Poisson distributions with D ~Poisson P tp tp tp where tp denotes the mortality rate which is modeled as a logarithm function of three sets of random effect as follows: ln. tp t p tp Random terms t, p, tp are temporal, spatial and spatiotemporal random effects respectively. Model hierarchy is completed by assuming prior distributions for each random vector α 1,..., t,..., T, φ 1,..., p,..., P, δ 11,..., tp,..., TP. For all these set of parameters, an Intrinsic Conditional AutoRegressive (ICAR) model is assumed, that can be expressed as multivariate Normal distributions parameterized in terms of mean vectors and precision matrices as follows: where matrices W and T 2 2 α ~MVN 0, M W φ ~MVN 0, M W P δ ~MVN 0, M W M W TP 2 W are adjacency matrices whose generic ij entry elements i and j are considered as neighbors, while wij 0 otherwise. Matrices M and wij 1 if M

11 are diagonal matrices with diagonal entries equal to the number of neighbors. As regards the temporal structure of the model, we consider only adjacent times as neighbors, i.e. w 1 iff tt' 1: this delivers a first order Random Walk for capturing temporal dynamics. As regards spatial random effects, provinces are considered as neighbors if they share a common boundary. The precision matrix of the spatiotemporal interaction term delta is specified as the Kronecker product of M W as suggested by Clayton (1996). As regards precision parameters 2, M W and 2 and 2, we assume independent small parameter Gamma as prior distribution both to preserve non informativeness and to take advantage of the conjugacy between the Normal and the Gamma distribution. In order to allow model identifiability, constraints are imposed on spatial ( φ ) and spatiotemporal ( δ ) random effects, while the unconstrained temporal random effects ( α ) capture the average log mortality rate at each year t Let θ α, φ, δ,,, Bayesian model can be written as follows: denote the set of model parameters. Eventually, the full θ D D θ α φ δ p p p p p p p p Where pθ D denotes the joint posterior distribution of model parameters. Since such distribution is not available in closed form, fully Bayesian inference proceeds by means of a MCMC algorithm which allows us to obtain samples from the posterior distribution. The MCMC algorithm consists in block Metropolis steps for sampling random effects 2 2 α, φ, δ. Gibbs steps can be adopted for sampling precision parameters,, 2, since their full conditional distributions are available in closed form. The algorithm we built performs an automatic tuning of the proposal distribution in order to achieve an acceptance rate of around 40%. With regard to forecasting, coherently with a Bayesian approach, predictions proceed on the basis of the posterior predictive distribution: a merit of the Bayesian approach is to properly propagate the uncertainty related to all model parameters. tt' 5. Applications Results shown in this section are based on posterior samples from the posterior distribution. High autocorrelation can be observed in the MCMC samples and convergence is quite slow. For this reason, after a 30,000 burn in period, we perform heavy thinning of the chains, building a final sample of 10,000 post convergence samples.

12 For the sake of brevity, only results referred to the female population are shown. In Figure 6, posterior means of the temporal random effects α are reported for each year and age class. The expected shape of the log mortality rate along ages is recovered from the model even if the proposed model does not explicitly take into account dependence between mortality rates by age. The downward shifting of the year specific curves testifies the decreasing trend in the mortality rates. Figure 6. Posterior means of the temporal random effects α vs age class. Each line corresponds to a year. j j-th age class Figure 7. Posterior means of the temporal random effects α (green lines) and provincespecific log rates in years 1992 and Year 1992 Year 2009 j j j-th age class j-th age class

13 As can be seen from Figure 7, while average levels of the log morality rates show a smooth trend over age classes, Province specific curves show quite a noisy behavior that can be ascribed to the lack of dependence between age classes in the estimated models. Figure 8. Posterior means of the spatial random effects φ for age classes 0 4 and Age class 0-4 Age class Figure 8 shows the average spatial structure at provincial level which was incorporated by the φ components. Figure 9. Model based estimates (black dots) and Bayesian credibility intervals (black lines). Direct estimates (red dots) and confidence intervals (red dashed lines). Year 1992 age class 0 4. Estimated Rate Province In Figure 9, model based estimates are compared with direct estimates: it can be seen that model based estimates are less variable than direct estimates. In fact, the confidence intervals which refer to direct estimates are on average 20% wider than the model based

14 credibility intervals. Such reduction in the uncertainty of the estimates can be attributed to the borrowing strength process. 6. Concluding Remarks In this preliminary paper, we discuss a spatio temporal Bayesian hierarchical model to estimate small area mortality rates at provincial level. Since provincial sub populations in Italy can widely vary, direct estimates of mortality rates can show huge uncertainty when considering specific age groups in some smaller provinces. In a preliminary descriptive analysis, we clearly demonstrated the existence of specific spatiotemporal patterns which can be easily incorporated in a comprhensive Bayesian hierarchical model. By exploiting the observed spatial association and temporal correlation trends, we proposed a model where provincial level mortality rates estimates can borrow strength from each other in order to reduce uncertainty related to the estimates. Model estimation is performed, under a Bayesian framework, by adopting an MCMC algorithm to obtain samples from the posterior distribution. As a result, it appears that model based estimates are less variable than direct estimates. Even if the dependence between mortality rates by age is not included in the model, the estimated log mortality rates assume a typical J shape curve which is totally coherent with the expected results. The ability of the proposed model in forecasting mortality rates needs to be evaluated; technically, predictions proceed on the basis of the posterior predictive distribution: prediction will be the next step of the present work, along with extensions that will allow to take account of age dependence. References Alho, J.M., Discussion of Lee (2000). North American Actuarial Journal 4, Brouhns N., Denuit M., Vermunt J. K., A Poisson log bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics 31 (2002) Brouhns N., Denuit M., Van Keilegom I., Bootstrapping the Poisson log bilinear model for mortality forecasting. Scandinavian Actuarial Journal Clayton D., Generalized linear mixed models. In Markov Chain Monte Carlo in Practice, ed. by W.R. Gilks, S. Richardson, and D.J. Spiegelhalter. Chapman and Hall,

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