Correction: The effects of increasing longevity and changing incidence on lifetime risk differentials: A decomposition approach

Abstract

[...]the fraction alive and healthy at age xi can be calculated by . Remaining lifetime risk at age 60 and lifetime risk decomposition for myocardial infarction, hip fracture and colorectal cancer, Sweden, males. https://doi.org/10.1371/journal.pone.0205550.g001 The incorrect summation index also appears in the S2 appendix. [...]there are errors in the S2 Appendix. [...]the formula writes as follows ... 푥 ≤ 푥 푖 ≤ 휔 휙 푥 푖 ,퐴 푙 푥 푖 ,퐴 휆 푥 푖 ,퐴 − ... 푥 ≤ 푥 푖 ≤ 휔 휙 푥 푖 ,퐵 푙 푥 푖 ,퐵 휆 푥 푖 ,퐵 = ... 푥 ≤ 푥 푖 ≤ 휔 ( 휙 푥 푖 ,퐴 − 휙 푥 푖 ,퐵 ) 4 푙 푥 푖 ,퐴 휆 푥 푖 ,퐴 + 푙 푥 푖 ,퐵 휆 푥 푖 ,퐵 3 + 푙 푥 푖 ,퐴 휆 푥 푖 ,퐵 + 푙 푥 푖 ,퐵 휆 푥 푖 ,퐴 6 5 ́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈ Contribution of Changing Incidence Risks + ... 푥 ≤ 푥 푖 ≤ 휔 ( 푙 푥 푖 ,퐴 − 푙 푥 푖 ,퐵 ) 4 휙 푥 푖 ,퐴 휆 푥 푖 ,퐴 + 휙 푥 푖 ,퐵 휆 푥 푖 ,퐵 3 + 휙 푥 푖 ,퐴 휆 푥 푖 ,퐵 + 휙 푥 푖 ,퐵 휆 푥 푖 ,퐴 6 5 ́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈ Contribution of Changing Survival Conditions (except Disease) + ... 푥 ≤ 푥 푖 ≤ 휔 ( 휆 푥 푖 ,퐴 − 휆 푥 푖 ,퐵 ) 4 휙 푥 푖 ,퐴 푙 푥 푖 ,퐴 + 휙 푥 푖 ,퐵 푙 푥 푖 ,퐵 3 + 휙 푥 푖 ,퐴 푙 푥 푖 ,퐵 + 휙 푥 푖 ,퐵 푙 푥 푖 ,퐴 6 5 ́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈ Contribution of Changing Disease Related Survival (3) -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Survival All Other Causes Incidence Survival MI Total Change Myocardial Infraction (MI), Sweden, Males, 2004-1994 Lifetime Risk Change in % Figure 1:

There are errors in the second and third sentences of the second paragraph of the Methods, as well as errors in the second and third equations. Please see the corrected sentences and equations here:

Hence, the fraction alive and healthy at age xi can be calculated by . The lifetime risk of becoming diseased from age x onward, lrx, can then be calculated by(2)where ω denotes the highest age attained. Eq 2 can be rewritten as(3)

For simplicity, we will write for and for .

The following changes occurred after the adjustment for the incorrect summation index:

* Myocardial infarction: slight increase in the lifetime risk estimates

* Colorectal cancer: slight increase in the lifetime risk estimates, the total change between the two lifetime risks compared, and the contributions of changes in disease incidence

* Hip fracture: slight increase in the lifetime risk estimates, the total change in the two lifetime risks compared, and the contribution of changes in survival; slight decline in the contribution of changes in disease incidence

These changes are represented in an updated Fig 1 and do not require any updates to the text. Please see the corrected Fig 1 here.

thumbnail

Download:

*

PPT

PowerPoint slide

*

PNG

larger image

*

TIFF

original image

Fig 1. Remaining lifetime risk at age 60 and lifetime risk decomposition for myocardial infarction, hip fracture and colorectal cancer, Sweden, males.

https://doi.org/10.1371/journal.pone.0205550.g001

The incorrect summation index also appears in the S2 appendix. Therefore, there are errors in the S2 Appendix. The correct S2 Appendix can be viewed below.

Supporting information

Showing 1/1: pone.0205550.s001.pdf

The effects of increasing longevity and changing

incidence on lifetime risk differentials: A

decomposition approach

(Appendix 2)

Marcus Ebeling

∗

, Karin Modig

†

, Anders Ahlbom

‡

and Roland Rau

§

Extending the decomposition by the disease related mortality

rate: equations and illustrative example

To investigate and illustrate the relationship between declining incidence of the respective dis-

ease and its role as a potential driver of increasing longevity over time, we incorporate disease-

related mortality as a third factor to the decomposition. If we are including the death rate of the

respective disease at age x,

푑

푥

, the rate of either dying or getting diagnosed at age x changes to

휇

푥

=

푚

푥

+

퐼

푥

+

푑

푥

,

(1)

where, for interpretation reasons,

푚

푥

now depicts the death rate for all other causes than the

disease in question. By applying this modification, the calculation of lifetime risk changes to

푙푟

푥

=

...

푥

≤

푥

푖

≤

휔

퐼

푥

푖

푒푥푝

[−

...

푥

≤

푦<푥

푖

퐼

푦

]

푒푥푝

[−

...

푥

≤

푦<푥

푖

푚

푦

]

푒푥푝

[−

...

푥

≤

푦<푥

푖

푑

푦

]

.

(2)

For simplicity, we will write again

휙

푥

푖

for

퐼

푥

푖

푒푥푝

[−

∑

푥

≤

푦<푥

푖

퐼

푦

]

and

푙

푥

푖

for

푒푥푝

[−

∑

푥

≤

푦<푥

푖

푚

푦

]

and,

furthermore,

휆

푥,푖

for

푒푥푝

[−

∑

푥

≤

푦<푥

푖

푑

푦

]

. Given these changes, the original decomposition formula

is extended by a third term, which indicates the contribution of changes in disease-specific

∗

University of Rostock & Max Planck Institute for Demographic Research, Rostock, Germany

†

Karolinska Institutet, Stockholm, Sweden

‡

Karolinska Institutet, Stockholm, Sweden

§

University of Rostock & Max Planck Institute for Demographic Research, Rostock, Germany

1

mortality between both populations. Accordingly, the formula writes as follows

...

푥

≤

푥

푖

≤

휔

휙

푥

푖

,퐴

푙

푥

푖

,퐴

휆

푥

푖

,퐴

−

...

푥

≤

푥

푖

≤

휔

휙

푥

푖

,퐵

푙

푥

푖

,퐵

휆

푥

푖

,퐵

=

...

푥

≤

푥

푖

≤

휔

(

휙

푥

푖

,퐴

−

휙

푥

푖

,퐵

)

4

푙

푥

푖

,퐴

휆

푥

푖

,퐴

+

푙

푥

푖

,퐵

휆

푥

푖

,퐵

3

+

푙

푥

푖

,퐴

휆

푥

푖

,퐵

+

푙

푥

푖

,퐵

휆

푥

푖

,퐴

6

5

́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈

Contribution of Changing Incidence Risks

+

...

푥

≤

푥

푖

≤

휔

(

푙

푥

푖

,퐴

−

푙

푥

푖

,퐵

)

4

휙

푥

푖

,퐴

휆

푥

푖

,퐴

+

휙

푥

푖

,퐵

휆

푥

푖

,퐵

3

+

휙

푥

푖

,퐴

휆

푥

푖

,퐵

+

휙

푥

푖

,퐵

휆

푥

푖

,퐴

6

5

́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈

Contribution of Changing Survival Conditions (except Disease)

+

...

푥

≤

푥

푖

≤

휔

(

휆

푥

푖

,퐴

−

휆

푥

푖

,퐵

)

4

휙

푥

푖

,퐴

푙

푥

푖

,퐴

+

휙

푥

푖

,퐵

푙

푥

푖

,퐵

3

+

휙

푥

푖

,퐴

푙

푥

푖

,퐵

+

휙

푥

푖

,퐵

푙

푥

푖

,퐴

6

5

́≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠Ø≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ ̈

Contribution of Changing Disease Related Survival

(3)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Survival

All Other

Causes

Incidence

Survival

MI

Total Change

Myocardial Infraction (MI),

Sweden, Males, 2004-1994

Lifetime Risk Change in %

Figure 1:

Decomposition of Lifetime Risk at Age 60 for Myocardial Infarction, Sweden,

Males, 1994-2004.

Fig. 1 shows an application of the three factor decomposition to the example of lifetime

risk for myocardial infarction for Swedish males, comparing the years 1994 and 2004. During

2

figshare

Download

Extending the decomposition by the disease related mortality rate: Equations and illustrative example.

(PDF)

S2 Appendix. Extending the decomposition by the disease related mortality rate: Equations and illustrative example.

https://doi.org/10.1371/journal.pone.0205550.s001

(PDF)

References

1. Ebeling M, Modig K, Ahlbom A, Rau R (2018) The effects of increasing longevity and changing incidence on lifetime risk differentials: A decomposition approach. PLoS ONE 13(4): e0195307. https://doi.org/10.1371/journal.pone.0195307 pmid:29672532