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Decomposing current mortality
differences into initial differences and
differences in trends: the contour
decomposition method
Dmitri A. Jdanov
Vladimir M. Shkolnikov
4th Human Mortality Database Symposium
Similarities and peculiarities on the way to longer life
Berlin, 22-23 May 2017
First stage: decomposition between two life expectancies. Discrete method for
decomposition of a difference between two life expectancies (Andreev, 1982, Arriaga,
1984, Pressat, 1985)
Second stage: general approaches which would solve the decomposition problem for
wider classes of demographic measures (Das Gupta, 1994, Das Gupta, 1999, Vaupel and
Canudas Romo, 2002, Shkolnikov, Valkonen, Begun and Andreev 2001, Andreev,
Shkolnikov and Begun 2002, Horiuchi, Wilmoth and Pletcher 2008).
All these methods solve one and same problem: estimation of effects on the change in the
dependent variable due to the change in each covariate. All these propose plausible
solution of the problem, but results may differ.
We use general stepwise replacement algorithm as baseline method. Our approach may
be repeated using other existing techniques but result will (probably) differ.
History
Decomposition of LE: USA vs. England and Wales
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0-4
5-9
10-1
415
-19
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
980
-84
85-8
990
-94
95-9
91
00
+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
USA: 1980 e(0)=70.0
E&W: 1980 e(0)=70.7
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
4
85-8
9
90-9
4
95-9
9
10
0+
Age group
USA: 2010 e(0)=76.3
E&W: 2010 e(0)=78.6
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0-4
5-9
10-1
415
-19
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
980
-84
85-8
990
-94
95-9
910
0+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
USA: 1980 e(0)=77.5
E&W: 1980 e(0)=76.8-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
4
85-8
9
90-9
4
95-9
9
100+
Age group
USA: 2010 e(0)=81.2
E&W: 2010 e(0)=82.6
Life expectancy, men
Life expectancy, women
1980 2010
Background
Assessment of contributions of different ages to the total difference at the time point T:
impact of every age depends on the difference in event rates at the first time point (t)
and on the temporal changes in these rates in each population between time points t
and T.
Core of the problem
The decomposition task is to split the final difference Δ𝐴𝐵= 𝑓 𝒎𝐴 − 𝑓 𝒎𝐵 into
additive age-specific cross-sectional (between country) contributions due to the initial
cross-sectional difference and the longitudinal (within country) contributions due to
event-rate changes.
The final cross-sectional age-components ∆𝐴𝐵𝑖 cannot be expressed via the
conventional age-components:
∆𝐴𝐵𝑖 ≠ ∆𝑎𝑏
𝑖 + (𝛿𝐴𝑎𝑖 − 𝛿𝐵𝑏
𝑖 ) .
Thus, the primary requirement is that at every age the sum of the initial and the trend
components is equal to the total age-specific component:
Δ𝐴𝐵 = (𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑖 + 𝑇𝑟𝑒𝑛𝑑𝑖)
𝑛
𝑖=1
= (Δ𝑎𝑏|𝐴𝐵𝑖 + 𝛿𝑎𝑏|𝐴𝐵
𝑖 )
𝑛
𝑖=1
,
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑖 + 𝑇𝑟𝑒𝑛𝑑𝑖 = Δ𝑎𝑏|𝐴𝐵𝑖 + 𝛿𝑎𝑏|𝐴𝐵
𝑖 = Δ𝐴𝐵𝑖 ,
𝑖 = 1, … , 𝑛
Stepwise replacement: one-dimensional problem
…
Initial stage
i = 1 i = 2 i = nA B
Δ𝐴𝐵1 Δ𝐴𝐵
Δ𝐴𝐵𝑛
Contour replacement
The sequence of element replacement in the four vectors in direction B→b→a→A
𝛿𝑏𝐵|𝐵1𝛿𝐴𝑎|𝐵1
Δ𝑎𝑏|𝐵1
…𝛿𝑏𝐵|𝐵 𝛿𝐴𝑎|𝐵
Δ𝑎𝑏|𝐵
𝛿𝑏𝐵|𝐵𝑛𝛿𝐴𝑎|𝐵𝑛
Δ𝑎𝑏|𝐵𝑛
Initial stage i = 1 i = 2 i = n
A B
a b
Δ𝐴𝐵1 Δ𝐴𝐵
Δ𝐴𝐵𝑛
Empirical example: USA and England and Wales, 1980-2010
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0-4
5-9
10-1
415
-19
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
980
-84
85-8
990
-94
95-9
910
0+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
Trend USA - Trend E&WInitial (1980)Final (2010)
USA: 1980 e(0)=70.0, 2010 e(0)=76.3
E&W: 1980 e(0)=70.7, 2010 e(0)=78.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
4
85-8
9
90-9
4
95-9
9
100+
Age group
Trend USA: 1980→2010
Trend E&W: 1980→2010
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0-4
5-9
10-1
415
-19
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
980
-84
85-8
990
-94
95-9
910
0+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
Trend USA - Trend E&WInitial (1980)Final (2010)
USA: 1980 e(0)=77.5, 2010 e(0)=81.2
E&W: 1980 e(0)=76.8, 2010 e(0)=82.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
4
85-8
9
90-9
4
95-9
9
100+
Age group
Trend USA: 1980→2010
Trend E&W: 1980→2010
Life expectancy, men
Life expectancy, women
Initial component:
-0.10
Trend Component:
-2.21
Conventional
Decomposition
in 2010:
-2.30
Initial component:
1.26
Trend Component:
-2.67
Conventional
Decomposition
in 2010:
-1.41
Causes of death
Stepwise replacement: two-dimensional problem (age + CoD)
…
Initial stage i = 1 i = 2 i = n
A B
Δ𝐴𝐵1 Δ𝐴𝐵
Δ𝐴𝐵𝑛
Contour decomposition by age and cause of death
Contour decomposition by age and cause of death
Empirical example: USA and England and Wales, 1980-2010
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0-4
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
Trend USA - Trend E&WInitial (1980)Final (2010)
USA: 1980 e(0)=70.0, 2010 e(0)=76.3
E&W: 1980 e(0)=70.7, 2010 e(0)=78.6
USA: 1980 e(0)=70.0, 2010 e(0)=76.3
E&W: 1980 e(0)=70.7, 2010 e(0)=78.6
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
InitialInitial
0.01 years-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA -Trend E&WTrend USA -Trend E&W
-2.21 years-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA -Trend E&W + Initial
-2.30 years
-1.1
-0.6
-0.1
0.4
0.9
1.4
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA
7.81 years-1.1
-0.6
-0.1
0.4
0.9
1.4
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend E&W
10.11 years
Life expectancy, men
Empirical example: USA and England and Wales, 1980-2010
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0-4
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
Trend USA - Trend E&WInitial (1980)Final (2010)
USA: 1980 e(0)=70.0, 2010 e(0)=76.3
E&W: 1980 e(0)=70.7, 2010 e(0)=78.6
USA: 1980 e(0)=70.0, 2010 e(0)=76.3
E&W: 1980 e(0)=70.7, 2010 e(0)=78.6
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Co
ntr
ibu
tio
n t
o e
(0)
dif
fere
nce
, ye
ars
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
InitialInitial
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA -Trend E&WTrend USA -Trend E&W
-2.67 years
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA -Trend E&W + Initial
-1.41 years
-1.1
-0.6
-0.1
0.4
0.9
1.4
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend USA
4.07 years
-1.1
-0.6
-0.1
0.4
0.9
1.4
0
5-9
10
-14
15
-19
20
-24
25
-29
30
-34
35
-39
40
-44
45
-49
50
-54
55
-59
60
-64
65
-69
70
-74
75
-79
80
-84
85
+
Age group
OtherExternal causes of deathOther cancersCa trachea, bronchus, lungCVD
Trend E&W
6.82 years
Life expectancy, women
References
Method (without CoD)
Jdanov, D. A.; Shkolnikov, V. M.; Van Raalte, A. A.; Andreev, E. M.:
Decomposing current mortality differences into initial differences and differences in
trends: the contour decomposition method
Demography (forthcoming).
Scripts (R)
Jdanov D.A. and Shkolnikov V.M. 2014. An R-script for the assessment of the
cross-sectional and the longitudinal components of a difference between two
values of an aggregate demographic measure by contour replacement. MPIDR
Technical Report TR-2014-003. Available at www.demogr.mpg.de
Thank you!
