Lung cancer mortality trends in Chile and six-year projections using Bayesian dynamic linear models

El objetivo fue analizar la tendencia de la tasa de mortalidad por cáncer de pulmón en Chile, durante el periodo 1990-2009 y proyectar estas tasas a seis años. La información de mortalidad fue obtenida del Ministerio de Salud de Chile. Para calcular las tasas se utilizaron las proyecciones de población según el Censo de 2002. Las tasas se estandarizaron usando la población mundial como referencia. Se ajustaron modelos lineales dinámicos bayesianos para estimar la tendencia entre 1990-2009 y proyectar el periodo 2010-2015. Durante el periodo se observa una reducción del 19,9% de la tasa de mortalidad en hombres, mientras que en mujeres, la tendencia es creciente con aumento de 28,4%. El modelo de segundo orden entregó un mejor ajuste en hombres y el de primer orden en mujeres. Entre 2010 y 2015, se mantiene la tendencia decreciente en hombres, en cambio se proyecta una estabilización en la tendencia de mortalidad por cáncer pulmonar en mujeres en Chile. Este tipo de análisis es útil para implementar sistemas de vigilancia epidemiológica y evaluar estrategias.


Abstract
The objectives were to analyze lung cancer mortality trends in Chile from 1990 to 2009, and to project the rates six years forward.Lung cancer mortality data were obtained from the Chilean Ministry of Health.To obtain mortality rates, population projections were used, based on the 2002 National Census.Rates were adjusted using the world standard population as reference.Bayesian dynamic linear models were fitted to estimate trends from 1990 to 2009 and to obtain projections for 2010-2015.During the period under study, there was a 19.9% reduction in the lung cancer mortality rate in men.In women, there was increase of 28.4%.The second-order model showed a better fit for men, and the firstorder model a better fit for women.Between 2010 and 2015 the downward trend continued in men, while a trend to stabilization was projected for lung cancer mortality in women in Chile.This analytical approach could be useful implement surveillance systems for chronic noncommunicable disease and to evaluate preventive strategies.

Introduction
Lung cancer accounts for 1.6 million deaths per year according to the World Health Organization (WHO) 1 .In Chile, cancer of the trachea, bronchi, and lungs (hereinafter "lung cancer") is the second cause of cancer mortality, following gastric cancer (Departamento de Estadísticas e Información en Salud, Ministerio de Salud.Mortalidad.http://www.deis.cl/?p=51, accessed on 23/ Jul/2013).
In developed countries, lung cancer incidence and mortality rates are declining in men and stabilizing in women thanks to changes in smoking prevalence.However, according to estimates in developing countries, incidence and mortality continue to increase due to endemic smoking prevalence 2 .
In Chile, the lung cancer mortality rate showed an upward trend in females and a slightly downward trend in males from 2001 to 2008 3 .The increase in the female population can be explained by the late adoption of smoking by women, due to sociocultural issues and exploitation of this characteristic by the tobacco industry 4 .
Time trend analysis of indicators like mortality is useful for monitoring a country's health status and the impact of health interventions 5 .Together with time trend analyses, it is useful to estimate the magnitude of the disease in the future, thus allowing optimization of resource allocation, services planning, and public policymaking 6 .
The current study aimed to analyze trends in the lung cancer mortality rate in men and women in Chile from 1990 to 2009 and to conduct projections for the next six years, using Bayesian dynamic linear models.

Time trend models
Standardized annual mortality rates were calculated by five-year age groups from 1990 to 2009 for men and women.The standardization method was direct adjustment, using the world standard as the reference population 8 .
The study used Bayesian dynamic linear models (DLM), where Y t represents the log of the mortality rate in time t and the model can be represented by the following system of equations: Where, F t is a vector of order P x 1 which is formed by co-variables, Ɵ t is the vector of the model's unknown parameters p, G t is a p x p order matrix that describes the trend of the parameters contained in Ɵ t over time, e t and w t represent random errors that are assumed to have typically normal distribution with mean 0, and variancecovariance matrices (depending on the structure of G t ) V t and W t respectively.
For the study's analyses, we specifically used two structures for F t and G t .
First-order model: Second-order model: What distinguishes DLM from ordinary timeseries models is the specification of the autocorrelation order for the model's structural parameters, making them more flexible.If there is a degree of autocorrelation, the second-order model is the most adequate.
Estimation of Bayesian DLM requires a stepwise filtering and smoothing process, in addition to defining a priori distributions for the unknown parameters that intervene in the models.When using the Bayesian paradigm in the estimation process, it is necessary to define knowledge a priori through distributions for the unknown parameters that intervene in the models.Thus, one assumes that the variances are independent and follow non-informative inverse gamma distributions.Additionally, one includes an a priori distribution for the series' baseline structural parameter, Ɵ o .The literature suggests specifying a normal distribution for the target parameter.Theoretical details on such models can be found in West & Harrison 9 and Gamerman & Lopes 10 .

Computational implementation
Adjustment of these models used the code's implementation in Winbugs 1.4.3 (http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml).Nevertheless, it is possible to use the dlm library from the R software package (http://www.rproject.org/),whose functions dlmModPoly and dlmModSeas are designed to estimate dynamic linear models 11 .To perform inference, chains of 60,000 iterations were generated, excluding the first 40,000 to eliminate the influence of baseline values and autocorrelation, which would allow ensuring the parameters' convergence in the MCMC strategy (Markov chain Monte Carlo stochastic simulation) used.Having estimated the models' parameters, one evaluates their goodness of fit using the deviance information criterion (DIC), the effective number of parameters (pD) 12 , the magnitude of the discrepancy observed a posteriori between the data expected by the models and the observed data (SCEp), and the model's predictive capacity based on the prediction error.
The model's projection was also compared to the data for 2010-2012, recently published by DEIS (http://www.deis.cl/?p=51, accessed on 23/Jul/2013).The study provides results of the best model, and based on which, projections of lung cancer mortality rates for the next six years in men and women in Chile, with the respective credible intervals.

Results
Time trend analysis of standardized lung cancer mortality rates in men and women in Chile from 1990 to 2009 showed a reduction of 19.9% in men, while in women there was an upward trend, namely a 28.4% increase (Figure 1).
The second-order DLM was the model with the best fit for the trend of standardized mortality rates in men, according to Bayesian goodness of fit criteria.Meanwhile, in women the first-order DLM showed the best fit (Table 1).Mortality data for the years 2010 to 2012 were used to obtain age-adjusted rates for both males and females, the value of which is contained in both models' confidence intervals (Table 2).Meanwhile, the differences between the expected and observed mortality rates were less than one (Table 2).
Based on each model, the study provided the six-year projections and credible intervals (Figures 2 and 3).In men, the downward trend observed in 1990-2009 was maintained.In women, the six-year projection showed stabilization of the lung cancer mortality rates, thereby interrupting the upward trend observed in the period under study.

Discussion
Chile is considered to have world-standard vital statistics 13 thanks to a systematic effort by the DEIS under the Ministry of Health of Chile in terms of coding and processing death certificates.Nevertheless, on-going efforts are needed to reduce the differences observed between urban and rural areas, counties, males and females, and age groups 14 .
One of the methods current employed to analyze trends in mortality rates is joinpoint regression 15 , useful for identifying changes in trends over time.The currently study used Bayesian DLM, which have been applied to a wide range of situations due to their predictive power, since the models' parameters are updated with each new observation, allowing them to adapt to the series' evolution and capturing changes in their behavior.Bayesian DLM are thus flexible for detecting change points and useful for forecasting 16 .
The downward trend in lung cancer in men has been described in developed countries, as has the upward trend in women, reflecting smok-ing prevalence at the population level 2 .In the case of women, a plateau effect has been seen following the upward trend, as described in the current study.
In Chile, the downward trend in men can be attributed not only to changes in smoking prevalence, but also to the longstanding exposure to arsenic in drinking water in the northern region of the country, followed by the installation of the first water treatment plants in 1971 17 .
One of study's limitations is that these rates are a summary measure for Chile as a whole and do not consider geographic differences, found in the relative risk of this disease due to the presence of arsenic in the north of the country 3 .Spatiotemporal studies are planned to take this phenomenon into account.
Another limitation is that the projections fail to consider changes in incidence of the disease or the incorporation of preventive strategies, so the results must be interpreted in the context of status quo..An aggravating factor is that Chilean schoolchildren (13 to 15 years of age) have the highest smoking rates in the Americas for their age group 20 .At the local level, studies on the general adult population in Santiago that evaluate the national trend reported 47% smoking prevalence in men and 27% in women in 1971.In 1984, the prevalence rates were 44% and 39%, respectively 21 .Finally, a study in 2003 found smoking prevalence rates of 46.5% in men and 39.4% in women 22 .
The current study used a methodology that allows modeling lung cancer mortality rates and producing projections.The methodology can be replicated to forecast mortality rates from other diseases in the population and make adjustments in public policies and implement surveillance systems for chronic non-communicable diseases and evaluate appropriate preventive strategies.Six-year projections according to first-order model and standardized lung cancer mortality rates in women.Chile 1990-2015.
Data source Data on lung cancer mortality [International Classification of Diseases, 9 th revision (ICD-9): 162 from the year 1990 to 1996 and C33-C34 from the 10 th revision (ICD-10): for the years 1997 to 2009], data were obtained from the available databases of the Department of Health Statistics and Information (DEIS) of the Ministry of Health of Chile, 1990-2009 (http://www.deis.cl/?p=51, accessed on 23/Jul/2013).Population data were obtained from projections based on the 2002 Census conducted by the Chilean National Institute of Statistics and the Latin American and Caribbean Demographic Center 7 .

ContributorsF.
Torres-Avilés participated in the study's conceptualization and design, adjustment of the models, data analysis and interpretation, writing of the article, and approval of the final version for publication.T. Moraga contributed on the data collection, adjustment of the models', data analysis and interpretation, writing of the article, and approval of the final version for publication.L. Núñez participated in the study's conceptualization, data analysis and interpretation, writing of the article, and approval of the final version for publication.G. Icaza colaborated on the study's conceptualization, data collection, analysis, and interpretation, writing of the article, and approval of the final version for publication.

Table 1
Selection criteria used in models for lung cancer mortality rates, Chile.