# Common Factor Cause-Specific Mortality Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Cause-of-Death Mortality

#### 2.1.1. Crude Mortality

#### 2.1.2. Net Mortality

#### 2.2. Competing Risk

#### 2.3. Model Estimation

#### 2.4. Forecast

**0**and covariance matrix ${H}^{c}$. To forecast ${K}^{c}$, we assumed the time series error to be none. From these forecasts, we in turn can derive the crude mortality forecasts for the Netherlands ${m}_{nl}^{c}(x,t)$ by applying (14).

#### 2.5. Old Ages

#### 2.6. Population Dynamics

- ${M}^{g}(x,t)$: net migration for males and females (g) for age x (from 0 to 99 years old) in year t. We assume migration for ages 100 to 120 to be zero;
- ${B}^{g}\left(t\right)$: total life births for males and females (g) in year t;
- ${m}_{nl}^{g}(x,t)$: the total crude mortality intensity for Dutch males and females (g) aged x in year t. This variable is obtained through Equation (3). We assumed mortality for individuals older than 120 in year t to equal that of a 120-year-old.
- ${E}^{g}(x,2016)$: the exposure for males and females on the first day of 2016, aged x (from 0 to 99 years old). We assume the first period exposure for ages above 99 to equal zero.

## 3. Data

## 4. Numerical Results

#### 4.1. Estimation

#### 4.1.1. Crude Mortality

#### 4.1.2. Net Mortality

#### 4.2. Forecast Results

#### 4.3. Population Dynamics

#### 4.3.1. Model Outcomes

**Independence copula and general effects**

**Respiratory disease shocks**

**Dependence**

#### 4.3.2. Model Comparison

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Frank and Clayton Copula Definition and Generator Function

## Appendix B. Maximum Likelihood Estimation of (24)

**Step 1:**

**Step 2:**

## Appendix C. Crude and Net Mortality for Males

**Figure A1.**Crude male mortality per age from historical data, for causes of death 1–6. A rainbow colour ramp is used to represent the course of years from red (1970) to violet (2015).

**Figure A2.**Comparison of the NL and EU crude (green) and net male mortality intensities from the Frank (red) and Clayton (blue) model. The solid lines represent 1970 and the dotted lines represent 2015.

**Figure A3.**Crude male mortality forecasts compared (Frank = dashed, Clayton = dotted). Age: 0 = yellow, 70 = blue, 80 = red, 85 = magenta, 90 = green.

**Figure A4.**The male log-mortality in 2050 for ages 0–90. Included are: the comparative model from Antonio et al. (2017) (black), the independence model (green), the single population model (red) and our model (blue). $\theta =2$ is assumed.

## Appendix D. Robustness Checks

**Table A1.**The resulting properties of forecasts using our cause-specific multi-population mortality model. Presented are the outcomes under different assumed dependence coefficients ($\theta $) and respiratory disease shocks (r). The included coefficients are: the dependency ratios in 2020, 2040 and 2060 (${d}_{2020},{d}_{2040},{d}_{2060}$), the period life expectancy for males and females in 2020 and 2060 ${}^{M}{l}_{2020}^{p}{,}^{M}{l}_{2060}^{p}{,}^{F}{l}_{2020}^{p}{,}^{F}{l}_{2060}^{p}$) and the male and female cohort life expectancy (${}^{M}{l}_{2020}^{c}{,}^{F}{l}_{2020}^{c}$).

Copula | r | ${\mathit{d}}_{2020}$ | ${\mathit{d}}_{2040}$ | ${\mathit{d}}_{2060}$ | ${}^{\mathit{M}}{\mathit{l}}_{2020}^{\mathit{p}}$ | ${}^{\mathit{M}}{\mathit{l}}_{2060}^{\mathit{p}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2020}^{\mathit{p}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2060}^{\mathit{p}}$ | ${}^{\mathit{M}}{\mathit{l}}_{2020}^{\mathit{c}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2020}^{\mathit{c}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

Independence | |||||||||||

- | 0.292 | 0.397 | 0.376 | 18.21 | 20.07 | 21.07 | 22.81 | 18.83 | 21.70 | ||

0.5 | 0.294 | 0.404 | 0.384 | 18.51 | 20.46 | 21.32 | 23.15 | 19.19 | 22.02 | ||

2 | 0.288 | 0.383 | 0.361 | 17.64 | 19.35 | 20.59 | 22.18 | 18.18 | 21.11 | ||

$\theta =0.5$ | |||||||||||

Frank | - | 0.292 | 0.397 | 0.377 | 18.22 | 20.15 | 21.20 | 22.92 | 18.86 | 21.81 | |

0.5 | 0.294 | 0.404 | 0.385 | 18.52 | 20.53 | 21.44 | 23.25 | 19.20 | 22.12 | ||

2 | 0.288 | 0.383 | 0.363 | 17.66 | 19.46 | 20.73 | 22.31 | 18.21 | 21.24 | ||

Clay | - | 0.295 | 0.408 | 0.395 | 18.63 | 21.16 | 23.01 | 24.49 | 19.41 | 23.38 | |

0.5 | 0.296 | 0.411 | 0.397 | 18.79 | 21.21 | 23.08 | 24.60 | 19.53 | 23.50 | ||

2 | 0.292 | 0.401 | 0.391 | 18.15 | 21.01 | 22.84 | 24.25 | 19.09 | 23.14 | ||

$\theta =1$ | |||||||||||

Frank | - | 0.292 | 0.398 | 0.379 | 18.25 | 20.25 | 21.31 | 23.05 | 18.89 | 21.93 | |

0.5 | 0.294 | 0.405 | 0.386 | 18.54 | 20.61 | 21.55 | 23.35 | 19.22 | 22.22 | ||

2 | 0.289 | 0.384 | 0.365 | 17.70 | 19.58 | 20.86 | 22.46 | 18.26 | 21.38 | ||

Clay | - | 0.294 | 0.402 | 0.385 | 18.45 | 20.57 | 21.98 | 23.75 | 19.10 | 22.55 | |

0.5 | 0.295 | 0.407 | 0.389 | 18.68 | 20.78 | 22.14 | 23.97 | 19.34 | 22.76 | ||

2 | 0.291 | 0.392 | 0.375 | 17.96 | 20.16 | 21.66 | 23.33 | 18.62 | 22.13 | ||

$\theta =2$ | |||||||||||

Frank | - | 0.293 | 0.400 | 0.383 | 18.36 | 20.46 | 21.58 | 23.36 | 19.01 | 22.22 | |

0.5 | 0.295 | 0.407 | 0.389 | 18.63 | 20.77 | 21.79 | 23.63 | 19.31 | 22.48 | ||

2 | 0.290 | 0.388 | 0.370 | 17.81 | 19.85 | 21.15 | 22.82 | 18.41 | 21.69 | ||

Clay | - | 0.293 | 0.400 | 0.383 | 18.38 | 20.50 | 21.47 | 23.23 | 19.03 | 22.08 | |

0.5 | 0.295 | 0.407 | 0.390 | 18.64 | 20.81 | 21.68 | 23.51 | 19.33 | 22.34 | ||

2 | 0.290 | 0.388 | 0.371 | 17.86 | 19.90 | 21.06 | 22.71 | 18.45 | 21.57 | ||

$\theta =5$ | |||||||||||

Frank | - | 0.298 | 0.416 | 0.399 | 19.16 | 21.25 | 22.53 | 24.27 | 19.79 | 23.19 | |

0.5 | 0.299 | 0.419 | 0.402 | 19.33 | 21.35 | 22.69 | 24.42 | 19.91 | 23.38 | ||

2 | 0.295 | 0.405 | 0.389 | 18.58 | 20.81 | 22.04 | 23.86 | 19.26 | 22.68 | ||

Clay | - | 0.293 | 0.399 | 0.380 | 18.34 | 20.28 | 21.21 | 22.97 | 18.96 | 21.84 | |

0.5 | 0.295 | 0.406 | 0.387 | 18.63 | 20.64 | 21.45 | 23.29 | 19.29 | 22.14 | ||

2 | 0.290 | 0.386 | 0.366 | 17.82 | 19.61 | 20.76 | 22.39 | 18.34 | 21.28 |

**Figure A5.**The observed period life expectancies for newborn, 45-year-old and 80-year-old males (blue) and females (pink) from our (solid) and the Antonio et al. (2017) (alternating) data set and the forecasts from Antonio et al. (2017) (black), the independence model (green), the single population model (red) and our model with $\theta =2$ (blue).

## Notes

1 | Source: Centraal Bureau voor de Statistiek. |

2 | For instance, the recent COVID-19 pandemic or an increase in the influenza virus-related deaths as observed by Actuarieel Genootschap (2018). |

3 | We refer to Enchev et al. (2017) for a survey and comparison of various multi-population models. |

4 | Enchev et al. (2017) analysed the ordinary Li–Lee model, two simplified versions of this model and the common age effect model of Kleinow (2015) and concluded that the regular Li–Lee model was the second-best performing model after the common age effect model. |

5 | Competing risks is the presence of censoring of the time of death from one cause in the event of death from another cause. |

6 | In addition to the clear extensions to the academic literature, we believe that the cause-specific extension is relevant for all practices which deal with longevity risk in the Netherlands and rely on the multi-population Antonio et al. (2017) model. |

7 | The LL model, as described in Antonio et al. (2017), is currently being used by Actuarieel Genootschap for the calculation of the Dutch life tables. |

8 | |

9 | The full definition of the survival functions as well as the first derivative and inverse of both corresponding generator functions are given in Appendix A, Equations (A1)–(A6). |

10 | Enchev et al. (2017) highlight the potential shortcomings in the re-estimation process of the time-dependent coefficient ${k}_{i}\left(t\right)$. They suggest a vector autoregressive model of order 1 (VAR(1)) instead of the AR(1) proposed by Li and Lee (2005). This is due to the sometimes diverging properties of the AR(1) model in a multi-population context. By the use of the VAR(1) model and its underlying covariance matrix, coherent forecasting can be retained. Moreover, this forecasting method does not significantly deviate from individual AR(1) forecasts (Enchev et al. 2017). |

11 | We acknowledge that we use a reduced variable set. A more comprehensive model could study all population flows with their own distinct functions, as can be seen in, e.g., Boumezoued et al. (2018). However, this is beyond the scope of our paper. |

12 | https://statline.cbs.nl/Statweb/ (accessed on 28 February 2019). |

13 | https://www.who.int/healthinfo/statistics/mortality_rawdata/en/ (accessed on 28 February 2019). |

14 | https://statline.cbs.nl/Statweb/ (accessed on 28 February 2019). |

15 | Keeping in mind the slim volume of cause-of-death data in many cases, we chose to include mortality numbers from Germany between 1970 and 1989, when the current country of Germany was split into the Federal Republic of Germany (West Germany/FRG) and the German Democratic Republic (East Germany/GDR). This is in contrast to the general mortality data used by Antonio et al. (2017). Moreover, in previous cause-of-death research by Arnold and Sherris (2015), mortality data of split Germany were not included, since no data were available for the German Democratic Republic before 1969. This does not pose a problem in our research, because the first year of our data set is 1970, and therefore, we accumulated the mortality knowledge of the German subsections to represent the total German mortality for the years preceding the fall of the Berlin Wall. |

16 | |

17 | For a full comprehensive display of the forecast, we refer to the illustrations in Figures S12–S15 of the Supplementary materials Section S.4. |

18 | This is not always the case, but results from our choice of dependence coefficient, as shown in Appendix D. |

19 | The sole difference is that in this case we have based the model on our acquired data set. |

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**Figure 1.**The levels of population/deaths for the Netherlands and the 14 selected European countries, for the data sets from this research (green) and Antonio et al. (2017) (red).

**Figure 2.**Cause-of-death mortality levels for males (blue) and females (pink) during the period 1970–2015.

**Figure 3.**Crude female mortality per age from historical data, for causes of death 1–6. A rainbow colour ramp is used to represent the course of years from red (1970) to violet (2015).

**Figure 4.**Comparison of the NL and EU crude (green) and net female mortality intensities from the Frank (red) and Clayton (blue) model. The solid lines represent 1970 and the dotted lines represent 2015.

**Figure 5.**Crude female mortality forecasts compared (Frank = dashed, Clayton = dotted). Age: 0 = yellow, 70 = blue, 80 = red, 85 = magenta, 90 = green.

**Figure 6.**The female log-mortality in 2050 for ages 0–90. Herein are included: the comparative model from Antonio et al. (2017) (black); the independence model (green); the single population model (red) and our model (blue). $\theta =2$ is assumed.

**Figure 7.**Descriptive illustrations of the population dynamics model under an assumed dependence variable $\theta =2$: (

**a**) Forecasted population composition: ages 0–20 (blue), 0–65 (green) total population (red). The dark solid line represents the CBS forecast from 2018 to 2060; (

**b**) dependency ratio changes for a multiplication of marginal mortality from respiratory diseases (cod 3) by factor 0.5 (green) and 2 (red) versus the base model (black). This is shown for Frank (dashed) and Clayton (dotted) specification.

**Figure 8.**The observed period life expectancies for 65-year-old males (blue) and females (pink) from our data set (solid), those of Antonio et al. (2017) (alternating) and CBS (dotted) and the forecasts from Antonio et al. (2017) (black), the independence model (green), the single population model (red) and our model with $\theta =2$ (blue).

**Table 1.**Contribution to overall mortality per cause-of-death, for males and females in the EU and NL (1970–2015).

EU | NL | ||||
---|---|---|---|---|---|

Code | Causes of Death | Male (%) | Female * (%) | Male (%) | Female (%) |

1 | Circulatory system | 32.1 | 37.0 | 29.6 | 31.1 |

2 | Cancer | 20.8 | 17.1 | 24.6 | 20.9 |

3 | Respiratory system | 4.9 | 4.2 | 7.4 | 6.2 |

4 | External causes | 5.8 | 3.8 | 4.2 | 3.2 |

5 | Infectious and parasitic diseases | 0.8 | 0.8 | 0.8 | 0.9 |

6 | Other | 35.6 | 37.2 | 33.4 | 37.7 |

**Table 2.**Schematic overview of the crude cause-specific mortality forecasts. ${m}_{x}^{i}$ represents the crude mortality forecast under copula i (we denote F for Frank and C for Clayton copula) and age x.

Gender | COD | Trend | Approx. | Model Comparison |
---|---|---|---|---|

Transition Age | (${\mathit{m}}^{\mathit{F}}$/${\mathit{m}}^{\mathit{C}}$) | |||

Male | 1 | - | - | = |

2 | +/− | 80–85 | ${m}^{F}<{m}^{C}$, ${m}_{90}^{F}>{m}_{90}^{C}$ | |

3 | - | - | ${m}^{F}<{m}^{C}$ | |

4 | - | - | ${m}^{F}<{m}^{C}$ | |

5 | +/− | 70–80 | ${m}^{F}<{m}^{C}$, ${m}_{90}^{F}>{m}_{90}^{C}$ | |

6 | - | - | = | |

Female | 1 | - | - | ${m}^{F}>{m}^{C}$ |

2 | - | - | ${m}^{F}>{m}^{C}$ | |

3 | = | - | ${m}^{F}<{m}^{C}$ | |

4 | - | - | ${m}_{90}^{F}<{m}_{90}^{C}$, ${m}_{80}^{F}>{m}_{80}^{C}$ | |

5 | +/− | 70–80 | ${m}^{F}<{m}^{C}$ | |

6 | - | - | = |

**Table 3.**Resulting properties of forecasts using our cause-specific multi-population mortality model. Presented here are the outcomes under different assumed dependence coefficients ($\theta $) and respiratory disease shocks (r). The included coefficients are: the dependency ratio in 2020, 2040 and 2060 (${d}_{2020},{d}_{2040},{d}_{2060}$), the period life expectancy for males and females in 2020 and 2060 ${}^{M}{l}_{2020}^{p}{,}^{M}{l}_{2060}^{p}{,}^{F}{l}_{2020}^{p}{,}^{F}{l}_{2060}^{p}$) and the male and female cohort life expectancy (${}^{M}{l}_{2020}^{c}{,}^{F}{l}_{2020}^{c}$). In the table, the differences ($\Delta $) are presented with respect to the independence copula model.

Copula | r | ${\mathit{d}}_{2020}$ | ${\mathit{d}}_{2040}$ | ${\mathit{d}}_{2060}$ | ${}^{\mathit{M}}{\mathit{l}}_{2020}^{\mathit{p}}$ | ${}^{\mathit{M}}{\mathit{l}}_{2060}^{\mathit{p}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2020}^{\mathit{p}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2060}^{\mathit{p}}$ | ${}^{\mathit{M}}{\mathit{l}}_{2020}^{\mathit{c}}$ | ${}^{\mathit{F}}{\mathit{l}}_{2020}^{\mathit{c}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

Independence | |||||||||||

- | 0.292 | 0.397 | 0.376 | 18.21 | 20.07 | 21.07 | 22.81 | 18.83 | 21.70 | ||

0.5 | 0.294 | 0.404 | 0.384 | 18.51 | 20.46 | 21.32 | 23.15 | 19.19 | 22.02 | ||

2 | 0.288 | 0.383 | 0.361 | 17.64 | 19.35 | 20.59 | 22.18 | 18.18 | 21.11 | ||

$\Delta $ | $\Delta $ | $\Delta $ | |||||||||

$\theta =0.5$ | |||||||||||

Frank | - | 0.000 | 0.000 | 0.001 | 0.01 | 0.08 | 0.13 | 0.11 | 0.03 | 0.11 | |

0.5 | 0.000 | 0.000 | 0.001 | 0.01 | 0.07 | 0.12 | 0.10 | 0.01 | 0.10 | ||

2 | 0.000 | 0.000 | 0.002 | 0.02 | 0.11 | 0.14 | 0.13 | 0.03 | 0.13 | ||

Clay | - | 0.003 | 0.011 | 0.019 | 0.42 | 1.09 | 1.94 | 1.68 | 0.58 | 1.68 | |

0.5 | 0.002 | 0.007 | 0.013 | 0.28 | 0.75 | 1.76 | 1.45 | 0.34 | 1.48 | ||

2 | 0.004 | 0.018 | 0.03 | 0.51 | 1.66 | 2.25 | 2.07 | 0.91 | 2.03 | ||

$\theta =1$ | |||||||||||

Frank | - | 0.000 | 0.001 | 0.003 | 0.04 | 0.18 | 0.24 | 0.24 | 0.06 | 0.23 | |

0.5 | 0.000 | 0.001 | 0.002 | 0.03 | 0.15 | 0.23 | 0.20 | 0.03 | 0.20 | ||

2 | 0.001 | 0.001 | 0.004 | 0.06 | 0.23 | 0.27 | 0.28 | 0.08 | 0.27 | ||

Clay | - | 0.002 | 0.005 | 0.009 | 0.24 | 0.50 | 0.91 | 0.94 | 0.27 | 0.85 | |

0.5 | 0.001 | 0.003 | 0.005 | 0.17 | 0.32 | 0.82 | 0.82 | 0.15 | 0.74 | ||

2 | 0.003 | 0.009 | 0.014 | 0.32 | 0.81 | 1.07 | 1.15 | 0.44 | 1.02 | ||

$\theta =2$ | |||||||||||

Frank | - | 0.001 | 0.003 | 0.007 | 0.15 | 0.39 | 0.51 | 0.55 | 0.18 | 0.52 | |

0.5 | 0.001 | 0.003 | 0.005 | 0.12 | 0.31 | 0.47 | 0.48 | 0.12 | 0.46 | ||

2 | 0.002 | 0.005 | 0.009 | 0.17 | 0.50 | 0.56 | 0.64 | 0.23 | 0.58 | ||

Clay | - | 0.001 | 0.003 | 0.007 | 0.17 | 0.43 | 0.40 | 0.42 | 0.20 | 0.38 | |

0.5 | 0.001 | 0.003 | 0.006 | 0.13 | 0.35 | 0.36 | 0.36 | 0.14 | 0.32 | ||

2 | 0.002 | 0.005 | 0.01 | 0.22 | 0.55 | 0.47 | 0.53 | 0.27 | 0.46 | ||

$\theta =5$ | |||||||||||

Frank | - | 0.006 | 0.019 | 0.023 | 0.95 | 1.18 | 1.46 | 1.46 | 0.96 | 1.49 | |

0.5 | 0.005 | 0.015 | 0.018 | 0.82 | 0.89 | 1.37 | 1.27 | 0.72 | 1.36 | ||

2 | 0.007 | 0.022 | 0.028 | 0.94 | 1.46 | 1.45 | 1.68 | 1.08 | 1.57 | ||

Clay | - | 0.001 | 0.002 | 0.004 | 0.13 | 0.21 | 0.14 | 0.16 | 0.13 | 0.14 | |

0.5 | 0.001 | 0.002 | 0.003 | 0.12 | 0.18 | 0.13 | 0.14 | 0.10 | 0.12 | ||

2 | 0.002 | 0.003 | 0.005 | 0.18 | 0.26 | 0.17 | 0.21 | 0.16 | 0.17 |

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**MDPI and ACS Style**

Zittersteyn, G.; Alonso-García, J.
Common Factor Cause-Specific Mortality Model. *Risks* **2021**, *9*, 221.
https://doi.org/10.3390/risks9120221

**AMA Style**

Zittersteyn G, Alonso-García J.
Common Factor Cause-Specific Mortality Model. *Risks*. 2021; 9(12):221.
https://doi.org/10.3390/risks9120221

**Chicago/Turabian Style**

Zittersteyn, Geert, and Jennifer Alonso-García.
2021. "Common Factor Cause-Specific Mortality Model" *Risks* 9, no. 12: 221.
https://doi.org/10.3390/risks9120221