SLIDE 1 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION
Arni S.R. Srinivasa Rao
Augusta University 1120 15th Street Augusta, GA 30912, USA Email: arrao@augusta..edu
James R. Carey
Department of Entomology University of California, Davis, CA 95616 USA and Center for the Economics and Demography of Aging University of California, Berkeley, CA 94720 Email: jrcarey@ucdavis.edu
- Abstract. Carey’s Equality arises while studying the average time left to die after forming a captive
cohort of individuals in a stationary population. This equality was first time observed in experimental
- entomology. We extend the ideas of Carey’s Equality, which arises in stationary populations to two
dimensions with the aim to answer questions related to aging, age-structure of the captive cohort. These ideas found to be useful for identifying individuals by age group in a captive cohort. Key words: Captive cohort, population age-structure, partition function. MSC: 92D25 Contents 1. Introduction 2 2. Structure of the 2-Dimensional Captive Cohort 3 3. Captive population Age-structure, Truncation and Partition Functions 5 3.1. Time left for a captive cohort and right truncation 6 4. Discussion 11 5. Conclusions 11 References 11
1
SLIDE 2 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 2
Although one of the basic canons of population studies is that knowledge of age composition is fundamental to the understanding the dynamics of a population, information on the age structure of a natural population is usually extremely limited. Although there are some long-term animals studies in which most or even all newborn are marked at birth and thus are of known age throughout a study[14], investigations with this information are the exception rather than the rule because of costs and
- logistics. Similar challenges exist for studies of natural arthropod populations where various methods
are used to estimate the age of individuals including, for example, mechanical damage, biochemical assays, physiological markers, and gene expression[15]. Because of the disconnect between the need for information on population age structure but the technical and budgetary challenges of obtaining the age of individuals for estimating age structure, James R. Carey and his colleagues developed a new approach to obtaining information about the age structure of wild populations that involved neither mark-recapture or age-estimating technologies. Rather the method was based on the concept that the death patterns observed in captured or marked individuals of unknown age would yield information on the age structure of the population from which these individuals were sampled. The starting point for this concept was to consider a closed stationary (zero growth) population in which the pre- and post-capture actuarial environments were identical and from which individuals were sampled at random and monitored through death. Using a simple table (See Table 1 in [6]) Carey showed that, under the stationarity assumption, that the post-capture death distribution of the randomly sampled individuals equals the age structure of the population. In this same paper Müller and his colleagues expressed this relationship analytically. See [1, 2] for a historical perspective on Carey’s Equality and related new developments. When population is stable, then the age-structure
- f the population is sensitive to the momentum [5].
Carey’s Equality[3] and a generalized theorem on Carey’s Equality[4] are true when the population growth is stationary. This equality is obtained from the information on age of an individual and life left at the time of capture of individual while forming a captive cohort in various experiments in ecology and population biology (for example, see [6, 7]). In one dimensional Carey’s Equality, f1(a) = f2(a) is true for any arbitrary age a of a population, where f1 and f2 are probability density functions of age composition and distribution of remaining lifespans in a stationary population [3]. This result is fundamental in establishing the experimental observations by Carey [6, 7]. f1(a) = f2(a) and other results describing the symmetries of life lived and remaining lifespans in [8], can be obtained as a direct consequences and applications of classical renewal theory frameworks [9, 10, 11, 12]. A generalized theorem of one dimensional Carey’s Equality was proved for sufficiently large stationary population without standard life table and renewal theory framework [4]. It was shown, in this generalized framework, that, there exists a graph from the family of exhaustive graphs constructed based on the co-ordinates of ages of individuals and their corresponding lives lived, will be equivalent to the graph obtained by the sequence of co-ordinates of the ages and corresponding lengths of life remaining (or life left) for each individual in a captive cohort [4]. All these above results are helpful in explaining the pattern of remaining lifespans for certain captive cohorts observed by Carey. Less
SLIDE 3 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 3
is known on determining an age structure of a captive cohort which is unknown at the time of formation, which will otherwise be helpful in understanding age wise or age-group wise remaining
- lifespans. Understanding age structure in a captive cohort formed for ecological and biodemographic
experiments could be very helpful in epidemiology studies for understanding transmission dynamics
- f parasites in some vector borne diseases.
The purpose of the current paper is to introduce a variation on Carey’s Equality in which the post-capture period of monitoring is truncated to less than the maximal age of individuals sampled in the hypothetical population. We introduce and develop concepts of two dimensional Carey’s Equality by considering age structure and proportion of population continuous in a range of entire age interval. We use these concepts for constructing partial age structure of the captured population which is unknown. We also consider truncation from right during the follow-up of captive cohort as an additional variable and use this variable in the partition function constructed in the process.
- 2. Structure of the 2-Dimensional Captive Cohort
Let x(a, i) be the ith−individual who was captured at age a and y(a, i) be the life left for ith−individual who was captured at age a. Then we represent the total number of individuals captured at age a as ´ ∞
0 x (a, i) di, who will have life left as
´ ∞
0 y (a, i) di units. Due to aging,
´ ∞
0 x (a, i) di
will become ´ ∞
0 x (b, j) dj during the time period b−a (b > a). Suppose on the positive quadrant, we
plot A, B, Q, and P where A = (a, 0) , B = (b, 0) , Q =
´ ∞
0 x (b, j) dj
´ ∞
0 x (a, i) di
respectively (see Figure 2.1), then the area of the quadrilateral, ABQ′P, say, K, for BQ′ > BQ, satisfies the inequality (2.1), K > 1 2 (b − a) ˆ ∞ x (a, i) di + ˆ ∞ x (b, j) dj
and the death rate per b − a time units is, 1 (b − a) ˆ ∞ x (a, i) di − ˆ ∞ x (b, j) dj
If death rate is zero, then K becomes, K = (b − a) ˆ ∞ x (b, j) dj (2.3) In the event that a captive cohort is formed from a stationary population at time t0, then ´ ∞
0 x (a, i, t0) di is a part or a sub-cohort of the captive cohort whose size is
´ ∞ ´ ∞
0 x (s, i, t0) dids
(here, s is a variable representing age). The sub-cohort ´ ∞
0 x (a, i) di will live for a
´ ∞
0 x (a, i, t0) di
time units if for each individual one dimensional Carey’s Equality explained in [3, 4] is true. Then the relation between each age in a sub-cohort of a captive cohort, size of the sub-cohort and life lived by individuals in the sub-cohort is given in
SLIDE 4 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 4
Figure 2.1. Aging process in a captive cohort a = ´ ∞
0 y (a, i) di
´ ∞
0 x (a, i) di
(2.4) The time left for a captive cohort describes the collective time that all the sub-cohorts of individuals
- f each age would live in the future. The time left for
´ ∞ ´ ∞
0 x (s, i, t0) dids is
´ ∞ ´ ∞
0 y (s, i) dids,
and if a captive cohort consists of individuals of only one age, a, then the total time left for the captive cohort is also a. We define a function of age-structure in the captive cohort, H1, and an associated set, A, such that, H1 : (0, ∞) → J(k) (2.5) where J(k) = {1, 2, · · · , k} for sufficiently large k, and, A = {(a, H1(a)) ∀a ∈ (0, ∞)} (2.6) where H1(a) = ´ ∞
0 x(a, i)di ∈ J(k). If H1 is constant, then, the captive cohort is uniformly distributed
- ver all ages or captive cohort consists of individuals of same age. If H1 is not constant, then, there
are three possibilities: (i) H1(a) > H1(a + h) for all h > 0, (ii) H1(a) < H1(a + h) for all h > 0, (iii) H1 is neither increasing nor decreasing for 0 < a < ∞. We define a function of life left in a captive cohort, H2, as follows: H2 : (t1, t2, · · · , t∞) → J(k) (2.7) for
SLIDE 5 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 5
(2.8) tm = mt1 ∈ (am−1, am) for m = 1, 2, · · ·∞, t∞ = a∞ and (2.9) inf (am−1, ∞) = am−1 for m = 1, 2, · · ·∞. The size of the captive cohort will start changing over the time t1, t2, · · · . The set C is computed as, C = {C(tm) : 1 ≤ m < ∞} (2.10) describe the collection of individuals from the captive cohort who are available at various future time
- points. Here C(tm) is number of individuals from the original captive cohort (say at time t0), who
are surviving at time tm, is given by, C(tm) = ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di
We assume, C(t∞) = 0.
- 3. Captive population Age-structure, Truncation and Partition Functions
With the help of the properties of H1 and H2 described in this section, we brought an expression for the number of individuals from the original captive cohort who are surviving at some time in the
- future. An expression to compute the time left for the captive cohort for varying truncation times,
tm (say) for m = 1, 2, ... is derived. We have showed that we can construct partial age-structure of the captive cohort up to certain age (say, as for as < tm) by using the sequences of time lived and time left up to tm. Theorem 3.1. C(tm), for m = 1, 2, ... is decreasing.
ˆ ∞ ˆ ∞ x (s, i) dids − ˆ ∞ x(a0, i)di > ˆ ∞ ˆ ∞ x (s, i) dids −
1
ˆ ∞ x(as, i)di and in general, ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di > ˆ ∞ ˆ ∞ x (s, i) dids −
m
ˆ ∞ x(as, i)di for each m = 1, 2, · · · , ∞. (3.1)
SLIDE 6 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 6
Age a0 a1 · · · a∞ Size ´ ∞
0 x(a0, i)di
´ ∞
0 x(a1, i)di
· · · ´ ∞
0 x(a∞, i)di
Table 1. Age-distribution in the captive cohort at t0. We will obtain, C(t1) > C(t2) > ...
- Remark 3.2. (i) Suppose, H1 is decreasing, then, for a0, a1, · · · , a∞ defined in (2.9) we have,
ˆ ∞ x(a0, i)di > ˆ ∞ x(a1, i)di > · · · > ˆ ∞ x(a∞, i)di = ⇒ ˆ ∞ ˆ ∞ x (s, i) dids − ˆ ∞ x(a0, i)di < ˆ ∞ ˆ ∞ x (s, i) dids − ˆ ∞ x(a1, i)di < · · · Remark 3.3. Suppose, C(tm) is decreasing, then we have, ˆ ∞ ˆ ∞ x (s, i) dids − ˆ ∞ x(a0, i)di > ˆ ∞ ˆ ∞ x (s, i) dids −
1
ˆ ∞ x(as, i)di > ˆ ∞ ˆ ∞ x (s, i) dids −
2
ˆ ∞ x(as, i)di . . . > ˆ ∞ ˆ ∞ x (s, i) dids − lim
m→∞ m
ˆ ∞ x(as, i)di By using the relationship in (2.4), we can write, C(tm), in terms of life left as follows: C(tm) = ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
1 as ˆ ∞ y(as, i)di
3.1. Time left for a captive cohort and right truncation. We derive a formula for computing the time left for the captive cohort at time tm, say, S(tm). Age distribution at cohort formation (say at t0) is given in Table 1. The time left for the captive cohort at t1, which was formed some time prior to t1 (say, at t0), is obtained as the difference between the time left at the time of formation of captive cohort and the time lived by captive individuals up to t1. Let, L(t1) be the life lived by the captive cohort up to t1 (including those lived up to a0 < t1), then,
SLIDE 7 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 7
Figure 3.1. Schematic structure of set A and collective life lived by A L(t1) = a0 ˆ ∞ x(a0, i)di + t1 ˆ ∞ x(a1, i)di + t1 ˆ ∞ x(a2, i)di + · · · + t1 ˆ ∞ x(a∞, i)di = a0 ˆ ∞ x(a0, i)di + t1 ∞
ˆ ∞ x(am, i)di
where tm ´ ∞
0 x(am, i)di is life lived by individuals whose age is am for m = 1, 2, · · ·∞. We will
- btain, S(t1), the time left by the captive cohort who are surviving at t1, as,
S(t1) = ˆ ∞ ˆ ∞ y (s, i) dids − L(t1). (3.4) Since a0 < t1 < a1, at t1, the individuals with age a1 would live for the duration (a1 − t1) and similarly, at t1, the individuals with age am would live for the duration (am−t1), for m = 2, 3, · · · , ∞. Hence, S(t1) = (a1 − t1) ˆ ∞ x(a1, i)di + (a2 − t1) ˆ ∞ x(a2, i)di + · · · + (a∞ − t1) ˆ x(a∞, i)di =
∞
(as − t1) ˆ ∞ x(as, i)di (3.5) Suppose, we truncate information from right after t1, then, the age structure that we are able to construct up to t1 is H1(a0). By (3.3), (3.4), and (3.5), we arrive at a relation between the total time
SLIDE 8 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 8
left for a captive cohort at t0 in terms of the time lived up to t1 by the captive cohort and the time left for the surviving individuals at t1 of the captive cohort as given in (3.6), ˆ ∞ ˆ ∞ y (s, i) dids = a0 ˆ ∞ x(a0, i)di + t1
∞
ˆ ∞ x(as, i)di +
∞
(as − t1) ˆ ∞ x(as, i)di. (3.6) Theorem 3.4. Suppose a captive cohort is formed at t0 and follow-up information is right truncated for the period (tm, ∞) for some tm > t0, then, the following relation hold: ˆ ∞ ˆ ∞ y (s, i) dids =
m−1
as ˆ ∞ x(as, i)di + ts
∞
ˆ ∞ x(as, i)di +
∞
(as − tm) ˆ ∞ x(as, i)di (3.7)
- Proof. The number of individuals alive at tm out of captive cohort are m−1
s=0
´ ∞
0 x(as, i)di and other
individuals have died at various stages before tm. Each sub-cohort of individuals whose age is as for s < m have lived collectively for the period m−1
s=0 as
´ ∞
0 x(as, i)di, then, using the arguments in
(3.3), L(tm), the life lived collectively up to tm by the sub-cohort of individuals of age as is given by, L(tm) = m−1
s=0 as
´ ∞
0 x(as, i)di,
if as < tm ts ∞
s=m
´ ∞
0 x(as, i)di,
if as > tm (3.8) Individuals from the captive cohort whose age is as > tm will live for the period (as−tm). By using the similar arguments of (3.5), we can compute, S(tm), the time left collectively for the sub-cohorts who are alive at tm, as, S(tm) =
∞
(as − tm) ˆ ∞ x(as, i)di for as > tm (3.9) Hence, by using (3.8) and (3.9), the time left for the captive cohort (which was formed at t0), i.e. ´ ∞ ´ ∞
0 y (s, i) dids is expressed as (3.7).
- Corollary 3.5. We can construct the age structure of the captive cohort up to age group as for
as < tm by using the sequence of time lived and time left up to tm in the Theorem 3.4. θm is the proportion of individuals in the captive cohort who are followed until death up to tm, is, θm = m−1
s=0
´ ∞
0 x(as, i)di
´ ∞ ´ ∞
0 x (s, i) dids .
We have, ´ ∞
0 x(a0, i)di ≤ θm
´ ∞ ´ ∞
0 x (s, i, t0) dids ≤
´ ∞ ´ ∞
0 x (s, i, t0) dids.
SLIDE 9 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 9
Suppose, H1 is increasing, then, we have, θ1 < θ2 < θ3 < · · · < θm < · · · . Information available during (tm, ∞) is not observed, so there are options for the death distribution of remaining individuals who are surviving at tm. Even-though, all the terms of the sequence (θm)m≥1 may not be possible to compute by tm, we can see that this sequence is convergent. The probability that the remaining people will survive at least for the period t1 (= (m + 1)t1 − mt1) after tm is one if (3.10) holds, else, there is a positive probability that the remaining cohort will finish within t1 units after tm. The information after tm which we are unable to follow-up, we denote by tm+b′ for b = 1, 2, · · · . ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di > ˆ ∞ x(am−1, i)di (3.10) For the post truncation period, the remaining individuals whose age is not determined will survive for at least t1 units after tm, if (3.10) holds. The number of individuals who are likely to die by tm+1′, say, D(tm+1′), are at least α1 and at most ´ ∞
0 x(am−1, i)di − 1 such that, the largest partition of the
integer ´ ∞ ´ ∞
0 x (s, i) dids − m−1 s=0
´ ∞
0 x(as, i)di − α1 is less than α1. Here, we have considered only
the partition of ´ ∞ ´ ∞
0 x (s, i) dids−m−1 s=0
´ ∞
0 x(as, i)di−α1 expressing as sum of distinct parts and
partitions such as (3.11) are not considered. ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di − α1 = 1 + 1 + · · · ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di − α1
= α1 + 1 + 1 + · · · ˆ ∞ ˆ ∞ x (s, i) dids −
m−1
ˆ ∞ x(as, i)di − 2α1
(3.11) For a general theory on integer partition, refer to [13]. When D(tm+1′) = ´ ∞
0 x(am−1, i)di − 1,
the slope of decline in H1 is relatively lower than that of d(tm+1′) = α1. Due to inverse relation between d(tm+1′) and loss of information after tm, the higher the value of D(tm+1′) in the interval
´ ∞
0 x(am−1, i)di − 1
- , the lesser is the loss of information due to truncation at tm. If D(tm+1′) =
´ ∞
0 x(am−1, i)di − 1, then, we will arrive at (3.12), and that ensures that the information loss after
the time tm+1′ is minimal. ˆ ∞ ˆ ∞ x (s, i) dids − m−1
ˆ ∞ x(as, i)di + D(tm+1′)
D(tm+1′) (3.12) q(tm+1′), the probability of dying of an individual during (tm, tm + t1) is, q(tm+1′) = D(tm+1′) ´ ∞ ´ ∞
0 x (s, i) dids − m−1 s=0
´ ∞
0 x(as, i)di
(3.13)
SLIDE 10
BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 10
Figure 3.2. Graphic showing the equality of the age distribution in a stationary population (upper) and the distribution of times-to-death of its members (lower). The implied age-specific mortality rates (qx) are 1/4, 1/6, 4/5 and 1.00 for age classes 0, 1, 2, and 3, respectively. Each segment labeled a, b, c or d in the top bar graphs corresponds to the fraction of the age class of the population that dies in the intervals 0, 1, 2, and 3 respectively (bottom bar graphs). For example, the percentage of the standing population that die in each segment labeled ‘a’ (top graph) sum to 40% in the time-to-death class 0 (bottom graphic). Or alternatively, as the population ages the number of individuals in segments a, b, c, and d in age class 0 (top graph) die off at times 0, 1, 2 and 3, respectively as shown in the time-to-death distribution (bottom graph). Source: Carey et al. (2012). If D(tm+1′) = α1, then, the inequality (3.12) do not hold and we need to continue steps that we did after arriving at the inequality (3.10). When (3.10) do not hold at the initial stage after truncation at tm, then, the arguments provided to arrive at inequality (3.12) hold. Note 3.6. When age-structure in the captive cohort does not follow any pattern then this situation can also be handled by using the technique of arranging the life lived in an ordered fashion explained for the one dimensional Carey’s Equality in [4]. One of the motivations for developing this framework of two-dimensional Carey’s Equality is due to absence of methods to estimate the age structure of wild insect populations. We wish attempt this by exploiting the information of lengths of life of captive individuals, population age distribution and time to death distribution. For example, from the pattern observed in Figure 3.2, we see the association of the population age distribution and the distribution of remaining time to death. Thus the former can estimated from knowledge of the latter.
SLIDE 11 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 11
Carey’s Equality with two variables is structured in this paper to suit analysis required to perform
- n a captive cohort when there exist very large number of individuals in each age-group of a popu-
- lation. Individuals in the captive cohort are constructed by life lived and size of the population in
each age-group. Unlike, in one dimensional Carey’s Equality [4], in this paper, the collective lengths
- f remaining lifespans associated with the age at death during follow-up (through the construction
- f two functions, H1 and H2) and number surviving at the time of follow-up (as explained through
Theorem 3.4 and Corollary 3.5) are simultaneously considered. Such considerations are new to the research in the Carey’s Equality which opened-up new doors in our theoretical understanding of the age-structure of insect captive cohorts. The theory proposed is sensitive to the proportion of the population at each age and can be modified to adapt, even if there is no monotonic pattern for the two functions, H1 and H2 introduced. Truncation from right is taken as a variable to construct par- tial age structure of the population. We have constructed partial age structure in a deterministic way without any probabilistic assumptions. However, we have given a formula for computing probability
- f dying of an individual who is alive after truncation, by partitioning the number of individuals
alive at the time of truncation. We have used distinct parts of the partition to suit monotonic nature
- f the age-structure and one can consider all the parts of the partition function when there is no
significant pattern of the age-structure. We have not used any probabilistic assumptions on the re- maining lifespans of the remaining individuals who are alive at the time of truncation in the captive cohort because such assumptions need to be justified and we have found no evidence in the published literature to start constructing partial age-structure. The method proposed in this paper can be also helpful in arranging age structure of a captive cohort on a continuous and flexible open intervals and geometric visualization of collective time lived and collective time remaining of a captive cohort.
We have essentially done following things in this article: (i) we have expressed captive cohort in a different way to previous works, (ii) each individual in every age is represented mathemtically in the formulations, (iii) sets of sub-cohots are analyzed using partition functions. Through this, we have achieved, (i) understanding internal structure of a captive cohort, (ii) construction of partial age-structure of captive cohort. References
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SLIDE 12 BEHAVIOR OF CAREY’S EQUALITY IN TWO-DIMENSIONS: AGE AND PROPORTION OF POPULATION 12
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