The Linear Link: Deriving Age-Specific Death Rates from Life Expectancy
Abstract
:1. Introduction
2. Data and Methods
2.1. Data
2.2. The Model
2.3. Algorithm
- Using the Kannisto mortality model (see Appendix A) extend to higher age groups up to age for all times t. The highest attainable age, , can be set for example to 120.
- Estimate the slope of the linear relation between life expectancy and the death-rates, , over the observation time t. This is done by using the method of the least squares approach, by minimizing the sum of squared residuals:Alternatively, the parameters of the model can be estimated by assuming that deaths follow a Poisson distribution (Brillinger 1986; Brouhns et al. 2002), , with . In order to use this approach death counts () and central exposure data () are needed. Sensitivity analysis shows that the difference between the two fitting procedure return minor discrepancies (see Appendix B in the Appendix for more details).
- Estimate the parameter by computing the singular value decomposition (SVD) of the matrix of regression residuals, , obtained in the previous step,and are matrices of left and right singular vectors, and is a diagonal matrix with singular values along the diagonal. The fist term of the , , is used for obtaining the estimates of . Parameter can be interpreted as the rate of mortality improvement over age.
- Smooth the and parameters using splines. This step is important to obtain graduated mortality curves and avoid projecting age-specific noise in the jump-off life table. However, if the graduation is not of interest or if the input data-set is large enough, this step can be skipped.
- Compute the initial mortality rates1 by , where .
- Optimize the mortality curve given in the previous step by finding the value of k where the difference between target life expectancy and an estimated life expectancy is below a tolerance level, for example 0.001, where represents the level of life expectancy at birth computed based on the mortality rates obtained in step (5). Usually k will be in the range of depending on the length of the forecast window.
3. Results and Illustration
4. Discussion
5. Reproducible Research
Author Contributions
Funding
Conflicts of Interest
Appendix A. The Kannisto Model
Appendix B. Maximum Likelihood Estimation
Appendix C. Rotation of Mortality Improvements
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1 | The change in age-specific death rates can be assumed to be constant over time, in which case the fitted is used in computing . Or, a shift in the speed of improvement can be imposed by “rotating” the coefficients. For more details see Section in the Appendix C. |
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Pascariu, M.D.; Basellini, U.; Aburto, J.M.; Canudas-Romo, V. The Linear Link: Deriving Age-Specific Death Rates from Life Expectancy. Risks 2020, 8, 109. https://doi.org/10.3390/risks8040109
Pascariu MD, Basellini U, Aburto JM, Canudas-Romo V. The Linear Link: Deriving Age-Specific Death Rates from Life Expectancy. Risks. 2020; 8(4):109. https://doi.org/10.3390/risks8040109
Chicago/Turabian StylePascariu, Marius D., Ugofilippo Basellini, José Manuel Aburto, and Vladimir Canudas-Romo. 2020. "The Linear Link: Deriving Age-Specific Death Rates from Life Expectancy" Risks 8, no. 4: 109. https://doi.org/10.3390/risks8040109