# The Linear Link: Deriving Age-Specific Death Rates from Life Expectancy

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## Abstract

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## 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 ${m}_{x,t}$ to higher age groups up to age $\omega $ for all times t. The highest attainable age, $\omega $, can be set for example to 120.
- Estimate the slope of the linear relation between life expectancy and the death-rates, ${\beta}_{x}$, over the observation time t. This is done by using the method of the least squares approach, by minimizing the sum of squared residuals:$$\sum _{x}{\left(\right)open="["\; close="]">log{m}_{x,t}-{\beta}_{x}log{e}_{\theta ,t}}^{}2.$$Alternatively, the parameters of the model can be estimated by assuming that deaths follow a Poisson distribution (Brillinger 1986; Brouhns et al. 2002), ${D}_{x}\backsim Poisson({E}_{x}^{c}\xb7{m}_{x,t})$, with ${m}_{x,t}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}exp({\beta}_{x}log{e}_{\theta}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\nu}_{x}k)$. In order to use this approach death counts (${D}_{x,t}$) and central exposure data (${E}_{x,t}^{c}$) 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 ${\nu}_{x}$ by computing the singular value decomposition (SVD) of the matrix of regression residuals, $\mathbf{R}$, obtained in the previous step,$$SVD\left[\mathbf{R}\right]={\mathbf{DPQ}}^{T}={d}_{1}{p}_{1}{q}_{1}^{T}+\dots ,$$$$\mathbf{R}=\left[\begin{array}{cccc}{\epsilon}_{0,1}& {\epsilon}_{0,2}& \cdots & {\epsilon}_{0,T}\\ {\epsilon}_{1,1}& {\epsilon}_{1,2}& \cdots & {\epsilon}_{1,T}\\ \vdots & \vdots & \ddots & \vdots \\ {\epsilon}_{\omega ,1}& {\epsilon}_{\omega ,2}& \cdots & {\epsilon}_{\omega ,T}\end{array}\right],$$$\mathbf{P}=[{p}_{1},{p}_{2},\dots ]$ and $\mathbf{Q}=[{q}_{1},{q}_{2},\dots ]$ are matrices of left and right singular vectors, and $\mathbf{D}$ is a diagonal matrix with singular values along the diagonal. The fist term of the $SVD$, ${d}_{1}{p}_{1}{q}_{1}^{T}$, is used for obtaining the estimates of ${\nu}_{x}$. Parameter ${\nu}_{x}$ can be interpreted as the rate of mortality improvement over age.
- Smooth the ${\beta}_{x}$ and ${\nu}_{x}$ 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 ${m}_{x,\tau}^{*}=exp\{{\beta}_{x}log{e}_{\theta ,\tau}^{*}+{\nu}_{x}k\}$, where $k=0$.
- Optimize the mortality curve given in the previous step by finding the value of k where the difference between target life expectancy ${e}_{\theta ,\tau}^{*}$ and an estimated life expectancy ${e}_{\theta ,\tau}$ is below a tolerance level, for example 0.001, where ${e}_{\theta ,\tau}$ 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 $\left(\right)$ 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

**Figure A2.**Comparison of the fitted mortality curves and parameter estimates of the Linear-Link model using the OLS+SVD and MLE fitting procedures. England & Wales female data for 1980–2018 period is used.

## Appendix C. Rotation of Mortality Improvements

**Figure A3.**Assumption of the change in ${\nu}_{x}$ pattern following the increase in life expectancy at birth from 75 to 102 years.

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**Figure 1.**Linear relation between life expectancy at birth and death-rates on a log-log scale, by age displayed together with the Pearson correlation coefficient, in the bottom left end of the panels. Each panel contains data for a specific age. The axis are labelled in normal scale for better interpretability. Based on HMD mortality data starting from 1980 for 41 countries and territories.

**Figure 2.**Linear relation between life expectancy at age 65 and death-rates on a log-log scale, by age displayed together with the Pearson correlation coefficient, in the bottom left end of the panels. Each panel contains data for a specific age. The axis are labelled in normal scale for better interpretability. Based on HMD mortality data starting from 1980 for 41 countries and territories.

**Figure 3.**Estimated parameters of the Linear-link model, using HMD data from 1980 to 2018 and life expectancy at birth $(\theta =0)$.

**Figure 4.**Observed and estimated death rates for female populations in 2018. Computed based on mortality data in the period 1965–1990.

**Figure 5.**Mean absolute errors (%) of the estimated log-death rates against the actual log-death rates between 1991 and 2018. Computed based on female mortality data in the period 1965–1990.

**Figure 6.**Relative errors (%) of the estimated log-death rates against the actual log-death rates between 1991 and 2018. Computed as (observed-estimated)/ observed log death rates. Based on female mortality data in the period 1965–1990.

**Figure 7.**Comparison of the mortality curves predicted by Lee-Carter and Linear-Link models in 2040 from female populations. The models are fitted on the 1980–2018 historical period.

1 | The change in age-specific death rates can be assumed to be constant over time, in which case the fitted ${\nu}_{x}$ is used in computing ${m}_{x}$. Or, a shift in the speed of improvement can be imposed by “rotating” the ${\nu}_{x}$ coefficients. For more details see Section in the Appendix C. |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Pascariu, 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