Symmetries Between Life Lived and Life Left in Finite
Stationary Populations
Francisco Villavicencio1,2
Tim Riffe3
[email protected]
[email protected]
1 Department
of Mathematics and Computer Science, University of Southern Denmark
Odense Center on the Biodemography of Aging
3 Max Planck Institute for Demographic Research, Rostock, Germany
2 Max-Planck
7th Demograhic Conference of “Young Demographers”
Prague, Czech Republic, February 11–12, 2016
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Overview
1
Introduction: The Brouard-Carey equality
2
Rao and Carey’s Theorem
Counterexample: Invalidity under continuous time
3
The Brouard-Carey equality for finite stationary populations in a
discrete-time framework
The concept of “death cohort”
Alternative Theorem
Proof
4
Conclusions
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Introduction: The Brouard-Carey equality
The symmetries between life lived and left in stationary populations
have drawn the attention of many scholars in the past years (Goldstein,
2009, 2012; Müller et al., 2004, 2007; Riffe, 2015; Vaupel, 2009).
“Carey’s equality”, as first coined by Vaupel (2009), establishes a
relationship between the age composition and the distribution of
remaining lifespans in stationary populations.
To our knowledge, this relationship was first found by Brouard (1989), and
was later and independently noticed by James Carey in the study of the
survival patterns of captive and follow-up cohorts of medflies.
=⇒ “Brouard-Carey equality”
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Introduction: The Brouard-Carey equality
In stationary populations of infinite size and continuous time (Brouard,
1989; Vaupel, 2009)
distribution of
remaining
years of life
−ω
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− x2
age
distribution
− x1
0
x1
Symmetries between life lived and left
x2
ω
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Introduction: The Brouard-Carey equality
This result has several applications in the study of human and non-human
populations with unknown ages.
Example
In capture-recapture studies where individuals are captured and then
followed until death, assuming stationarity, it can be inferred that:
unobserved distribution of capture ages
=
observed distribution of the follow-up durations
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Rao and Carey’s Theorem
Rao and Carey (2015) claim to have an alternative and innovative proof of
the Brouard-Carey equality that
1
It is inspired on the empirical observation of survival patterns in
captive med-flies; and
2
Provides a demonstration of the Brouard-Carey equality for finite
stationary populations.
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Rao and Carey’s Theorem
Theorem (Rao and Carey (2015))
Suppose (X , Y , Z ) is a triplet of column vectors, where X = [x1 , x2 , ..., xk ]T ,
Y = [y1 , y2 , ..., yk ]T , Z = [z1 , z2 , ..., zk ]T representing capture ages, follow-up durations,
and lengths of lives for k-subjects, respectively. Suppose, F(Z), the distribution function
of Z is known and follows a stationary population.
Let G1 be the graph connecting the co-ordinates of SY , the survival function whose
domain is N(k) = 1, 2, 3, ..., k, i.e. the set of first k positive integers and SY (j) = yj for
j = 1, 2, ..., k.
Let G2 be the graph connecting the co-ordinates of CX , the function of capture ages
whose domain is N(k) and CX (j) = xj for all j = 1, 2, ..., k. Suppose CX∗ (−j) = xj for all
j = 1, 2, ..., k.
Let H be the family of graphs constructed using the co-ordinates of CX∗ consisting of
each of the k! permutations of graphs.
Then one of the members of H (say, Hg ) is a vertical mirror image of G1 .
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Rao and Carey’s Theorem
Hg
capture ages (x)
follow−up durations (y)
G1
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
9
10
index
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Rao and Carey’s Theorem
In their proof, Rao and Carey assume that
∀yi ∈ Y there exists xj ∈ X such that yi = xj
In other words, that for every subject of the data set that has a follow-up
duration of yi there is an individual j whose capture age is exactly xj = yi .
Without this assumption, the theorem cannot be proved.
In fact, this assumption is wrong in continuous time!
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Rao and Carey’s Theorem: Counterexample
Definition (Stationary population)
A stationary population results from the continued operation of three
demographic conditions (Preston et al., 2001):
1
Age-specific death rates constant over time
2
Flow of births is constant over time
3
No migration
Proprieties:
Population size is constant
New individuals can only come via birth flow
#births = #deaths
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Rao and Carey’s Theorem: Counterexample
Counterexample: Stationary population where individuals within cohorts
have varying lifespans, but the exact same pairing of lifespans with
individuals is repeated in each year (more generous than only stationarity)
Individual
1
2
3
4
5
6
7
8
9
10
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Birth time
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
Lifespan
0.30
8.10
6.20
7.40
5.20
3.60
0.60
6.70
1.10
3.10
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Rao and Carey’s Theorem: Counterexample
Figure : Lexis diagram of a long series of birth cohorts
8
7
6
age
5
4
3
2
1
0
t−8
t−7
t−6
t−5
t−4
t−3
t−2
t−1
t
t+1
t+2
t+3
calendar time
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Table : Life lived (xj ) and life left (yj ) data from any census carried out in the
stationary part of the series at exact times t, t + 1, etc.
Individual
xj
yj
Individual
xj
yj
Individual
xj
yj
Individual
xj
yj
1
7.85
0.25
12
2.75
3.45
23
3.55
1.65
34
4.25
2.45
Villavicencio & Riffe
2
6.85
1.25
13
1.75
4.45
24
2.55
2.65
35
3.25
3.45
3
5.85
2.25
14
0.75
5.45
25
1.55
3.65
36
2.25
4.45
4
4.85
3.25
15
6.65
0.75
26
0.55
4.65
37
1.25
5.45
5
3.85
4.25
16
5.65
1.75
27
3.45
0.15
38
0.25
6.45
6
2.85
5.25
17
4.65
2.75
28
2.45
1.15
39
0.15
0.95
7
1.85
6.25
18
3.65
3.75
29
1.45
2.15
40
3.05
0.05
Symmetries between life lived and left
8
0.85
7.25
19
2.65
4.75
30
0.45
3.15
41
2.05
1.05
9
5.75
0.45
20
1.65
5.75
31
0.35
0.25
42
1.05
2.05
10
4.75
1.45
21
0.65
6.75
32
6.25
0.45
43
0.05
3.05
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3.75
2.45
22
4.55
0.65
33
5.25
1.45
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Rao and Carey’s Theorem: Counterexample
Therefore, in continuous time and a finite population for every subject
of the data set that has a follow-up duration of yi does not necessarily
exist an individual j whose capture age is exactly xj = yi .
This graph cannot be symmetric:
Hg
capture ages (x)
follow−up durations (y)
G1
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
9
=⇒ Theorem 1 by Rao and
Carey (2015) is invalid!
10
index
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The Discrete-time Framework
However, if we consider age and time intervals or classes, the result from
our example may change:
Table : Age and life left data from our example grouped in categories
Age / life left category
No. individuals in age
No. individuals in life left
[0, 1)
9
9
[1, 2)
7
7
[2, 3)
7
7
[3, 4)
7
7
[4, 5)
5
5
[5, 6)
4
4
[6, 7)
3
3
[7, 8)
1
1
If time is discretized in
Age intervals
Time intervals
then, the Brouard-Carey equality may hold.
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The Brouard-Carey equality in discrete-time framework
The concept of “death cohort”: Individuals dying in the same
time-interval.
3
3
2
2
1
1
0
t−1 B(t − 1)
t
B(t)
t+1 B(t + 1) t+2 B(t + 2) t+3
calendar time
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t−1 D(t − 1) t
D(t)
remaining−time (life left)
age (life lived)
Figure : Two Lexis diagrams of the same population with individuals grouped by
birth cohorts (left-hand side, classical Lexis diagram), and by death cohorts
(right-hand side)
0
t+1 D(t + 1) t+2 D(t + 2) t+3
calendar time
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The Brouard-Carey equality in discrete-time framework
Theorem (The Brouard-Carey equality in finite stationary populations)
Suppose there is a finite stationary population that is observed at regular time
points.
Suppose age and time are discretely measured, using the same time units for both.
Assume that the birth flow is not subject to stochastic variation, i.e., births are
equally distributed (but not necessarily uniformly distributed) within every
time-period between observations. Let’s define,
Nx (t): number of individuals in age [x, x + 1) at exact time t, and
Ωy (t): number of individuals with [y , y + 1) life left at exact time t.
Then,
Nx (t) = Ωy (t) for x = y and ∀t.
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(1)
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The Brouard-Carey equality in discrete-time framework
Lexis diagram of birth cohorts
Lexis diagram of death cohorts
2
N1(t)
N1(t + 1)
N1(t + 2)
Ω1(t − 1)
Ω1(t)
1
1
N0(t)
0
t−1
B
Ω0(t)
N0(t + 1)
D(t)
B
t
Ω0(t + 1)
t+1
t+2
t−1
calendar time
t
D(t+1)
t+1
remining−time (life left)
age (life lived)
2
0
t+2
calendar time
Nx (t) = Ωx (t), and in particular, N0 (t) = Ω0 (t), N1 (t) = Ω1 (t), etc.
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The Brouard-Carey equality: Proof
Let’s prove that
N0 (t) = Ω0 (t) ∀t
(2)
Note that N0 (t) = B − L D0 (t − 1) ∀t, where
L Dx (t − 1): individuals that born and died in [t − 1, t), that is,
deaths in the “lower-triangle” of the Lexis diagram.
The assumptions of the Theorem imply that L Dx (t) = L Dx ∀t (does not
depend on time). Therefore,
N0 (t) = B − L D0 ∀t
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The Brouard-Carey equality: Proof
N0 (t) = B − L D0 ∀t
(3)
age (life lived)
2
N1(t)
N1(t + 1)
N1(t + 2)
1
N0(t)
LD0
0
t−1
N0(t + 1)
LD0
B
B
t
t+1
t+2
calendar time
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The Brouard-Carey equality: Proof
On the other hand,
D(t) = Ω0 (t) + κ0 (t) ∀t
where
D(t): deaths of all ages that will occur in [t, t + 1),
Ω0 (t): individuals that at exact time t have [0, 1) years of life left, and
κ0 (t): new individuals that join the population in [t, t + 1) and that
die before t + 1.
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The Brouard-Carey equality: Proof
N0 (t) = B − L D0
D(t) = Ω0 (t) + κ0 (t)
2
N1(t)
N1(t + 1)
N1(t + 2)
Ω1(t − 1)
Ω1(t)
1
1
N0(t)
LD0
0
t−1
Ω0(t)
N0(t + 1)
B
t+1
t+2
t−1
calendar time
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κ0(t + 1)
D(t)
B
t
Ω0(t + 1)
κ0(t)
LD0
t
D(t+1)
t+1
remining−time (life left)
age (life lived)
2
0
t+2
calendar time
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The Brouard-Carey equality: Proof
Since the population size is constant,
D(t) = B ∀t
As new individuals can only come from births.
κ0 (t) = L D0
that is, newborns from [t, t + 1) that die before t + 1. As a result,
Ω0 (t) = D(t) − κ0 (t) = B − L D0 = N0 (t) ∀t.
Q.E.D
(for a complete proof, see Villavicencio & Riffe (in review)).
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Conclusions
The Brouard-Carey equality implies that the distribution of the capture ages
equals the distribution of the remaining lifespans.
Rao and Carey (2015) introduce a novel approach for finite stationary
populations, but the proof of their theorem stands invalid at this time.
Our main contribution is to offer a formal proof of the Brouard-Carey
equality in a discrete-time framework. The discretization of time provides a
more realistic set up, but makes the demonstrations more complex.
The Brouard-Carey equality is a useful identity when capture ages are
unknown, which is a common scenario in many field studies on wild animals.
It can also be applied to human populations when the ages of individuals are
unknown, as it may be the case in historical data or other kind of incomplete
demographic data.
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Thanks for your attention!
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References
Brouard, N. (1989). Mouvements et modèles de population. Yaoundé, Cameroon: Institut de
Formation et de Recherche Démographiques (IFORD).
Goldstein, J. R. (2009). Life lived equals life left in stationary populations. Demographic
Research 20 (2), 3–6.
Goldstein, J. R. (2012). Historical Addendum to ‘Life lived equals life left in stationary
populations’. Demographic Research 26 (7), 167–172.
Müller, H. G., J. L. Wang, J. R. Carey, E. P. Caswell-Chen, C. Chen, N. Papadopoulos, and
F. Yao (2004). Demographic window to aging in the wild: Constructing life tables and
estimating survival functions from marked individuals of unknown age. Aging Cell 3 (3),
125–131.
Müller, H. G., J. L. Wang, W. Yu, A. Delaigle, and J. R. Carey (2007). Survival and aging in
the wild via residual demography. Theoretical Population Biology 72, 513–522.
Preston, S. H., P. Heuveline, and M. Guillot (2001). Demography: Measuring and Modeling
Population Processes. Malden, MA, US: Blackwell Publishers.
Rao, A. S. R. S. and J. R. Carey (2015). Generalization of Carey’s equality and a theorem on
stationary population. Journal of Mathematical Biology 71 (3), 583–594.
Riffe, T. (2015). The force of mortality by years lived is the force of increment by years left in
stationary populations. Demographic Research 32 (29), 827–834.
Vaupel, J. W. (2009). Life lived and left: Carey’s equality. Demographic Research 20 (3), 7–10.
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