International Mathematical Forum, Vol. 12, 2017, no. 10, 469 - 479 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7225

Life Tables - Key Parameters and

Relationships between Them

Ts. P. Kovacheva

Technical University of Varna Department Mathematics 1, Studentska str., 9010 Varna, Bulgaria

Copyright © 2017 Ts. P. Kovacheva. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the paper is to make an overview of the emergence and use of the life tables in the analysis of the duration of a given generation. The key parameters and the relationship between them of the complete life tables are discussed. As an example are considered and compared two complete life tables based on gender for the Bulgaria and USA.

Keywords: life table, biometric characteristics, vital statistics, life expectancy, probability

1 Introduction

The life tables (biometric, actuarial, for life expectancy) are a basic tool of the vital statistics, insurance, different fields of health, facilitate comparisons between regions, etc. [2, 6]. They are used to compute life expectancy at birth and at different ages. The tables are also used to compute many other parameters: death probabilities, probabilities of survival between two ages, years of life lived and the number of survivors at different ages. The construction of a life table makes it possible to summarize mortality within a population at a given time (period life table). Their construction and development is associated with the development of statistics and probability theory. The mathematical probability theory originates from the middle of the XVII century with the work of the French mathematicians Blaise Pascal and Pierre de Fermat. In the second half of this century have appeared

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also the first works associated with mathematical calculations of life insurance. In his book, the English scientist John Graunt (1620-1674) developed for the first time statistical methods for population studies, which are the basis of the modern demographics. He made an in-depth analysis of the mortality of the population of London and explored the degree of impact of different diseases. Graunt constructed a life table, which contained data for the probabilities of survival and death to a certain age. The English astronomer, mathematician and physicist Edmond Halley (1656-1742) has analyzed in detail the data from a study of the registers of the town of Breslau (Wrocław) in the period 1697-1691 relative to the number of births and deaths in the town. The table of Halley indicated the mortality rate based on age. Abraham de Moivre (1667-1754) was a French mathematician who made a number of applied researches on the life tables. Based on the statistical data of Edmond Halley, he concluded that the mortality of the older generation is subject to continuous even distribution. The Swedish astronomer, statistician and secretary of the Royal Academy of Sciences Pehr Wargentin (1717-1783) developed in 1766 the first life table for the entire population of an individual country - Sweden. In 1760, the founder of statistical physics the Swiss mathematician and physicist Daniel Bernoulli (1700 - 1782) supplemented the life table model by creating a concept for differentiated life tables based on specific causes of death. In 1820, the English mathematician and actuary Benjamin Gompertz (1779-1865) offered a union of actuarial life tables to 17 English insurance companies. He demonstrated that the survival function of two persons was amended in arithmetic progression, and the difference of the values belonging to them - in geometric. Gompertz offered a "law of mortality", according to which the death is a consequence of two reasons: one based on the constancy of the intensity of mortality and reduction in the number of people who lived up to the age of x in geometric progression and the other – the decrease, proportionally in time, of the ability to withstand death. The graphical illustration of the generations’ aging has been presented for the first time by the prominent German statistician Wilhelm Lexis (1837-1914) and bears his name also nowadays. In 1944, during the construction of a life table, A. Coale accepted the life expectancy in the age band of 10 years as an indicator of the mortality rate. The probability of death was determined for each age group according to a method of linear regression. In 1948, Thomas Greville created patterns and algorithms for obtaining life tables considering the different reasons. In the same year, in England was founded the Institute for Actuaries, which initiated and managed the life tables necessary for the insurance. In 1955, experts from the UN, by statistical processing of 158 different life tables, constructed a series of 40 tables ordered by the increase of the average life expectancy from 18 to 75 years divided into groups and by mortality rate. For the intermediate ages, a parabolic regression of the probability of death was used. To improve the mortality model, along with the indicator life expectancy a number of other parameters were introduced. S. Ledermann and J. Bria (France) processed in 1959 a large set of life tables with the methods of factor analysis. As a result, 4 factors were separated, explaining 95% of the variance of the age probabilities of death.

Life tables - key parameters and relationships between them 471

Basing on these results, Jean Bourgeois-Pichat constructed a five-dimensional system of life tables, published in 1962 and 1966. In 1966, A. Coale and P. Demeny made a statistical survey of 326 life tables. They examined the age possibilities of death and their logarithms as functions of one parameter (average life expectancy in the age span of 5 years), measuring the overall mortality rate. In 1972 were created also the so-called type (model, template) life tables for populations with scarce or unreliable statistical information in order to assess the age mortality and the life expectancy of such populations. In the following years, a number of authors developed and advanced the model of aging of the so-called conditional (hypothetical) generation of the life tables as a chain of events in dynamic aggregates and derived the main parameters of this stochastic process. Based on these results solutions were sought for the impact on the parameters of the life tables.

2 Key parameters of the life tables

For the construction of a life table a given aggregate l0 of simultaneously born people is considered. Generally, l0 10 000, 100 000, 1 000 000 is accepted and it is called the root (radix) of the table. In the columns, the data for different demographic parameters (biometric characteristics) ordered by age is plotted [2, 4, 7]. Obtaining these parameters requires observation of the entire life cycle of an actual generation for 100 and more years. The resulting table of the actual generation will have historical value because it has lived in conditions other than the current. Thus, the study and the forecast of the process of the current generation’s reduction under the influence of the mortality is realized basing on a conditional generation. The life tables are constructed by real data on mortality of the simultaneously living generations during a specified calendar period. There are various methods for constructing life tables. The main difference between them consists in the selection of a basic parameter used for the calculation of the rest of the parameters. Most often, this parameter is the probability of dying qx between ages x and x 1 given by the statistical institutes on the base of a methodology adopted by them. These probabilities form the bases of the life table and all subsequent columns are derived from it. For each year x of the life of the people from the examined aggregate, the following parameters are determined [3, 4]:

 px - conditional probability of surviving between age x and age x 1 (probability of surviving between ages x and x 1) as

px  qx 1 (1)

px 1qx (2)

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 lx - number of the people who have lived up to the age of x (number surviving to age x )

lx1  lx px  lx 1 qx  (3) It is the first major aggregate of the living people, which presents the conditional generation and its decrease with increasing the age.

 d x - number of people who die in the interval x,x 1 (number of deaths)

dx  lx lx1  lxqx , lx  lx1 (4)

 Lx - number of person-years accumulated from the entire group in the interval x,x 1 or average number of people living (person-years lived between ages x and x 1 ) l  l 1 L  x x1  l  d (5) x 2 x 2 x This is true in the case of uniform (linear) decrease in the number of the people who lived up to a certain age interval. The given parameter is calculated for all ages except of the first (0 years) and the last year  . It appears to be the second major aggregate of the living people.

 Tx - number of person-years of the expected life of the person who has lived to the age of x (total number of person-years lived above age x )  Tx   Lt (6) tx This parameter characterizes the total number of person-years that the group of persons is remaining to live between age x and the limit age. 0  ex - average life expectancy (average expectation of life at age x , i.e. average number of years remaining to live for the group of persons who reached a certain age x

0 Tx ex  (7) lx 0 The first calculated value of ex indicates the average life expectancy for the relevant group.

 Rx - mortality rate

d x Rx  (8) Lx

 Px - parameter of survival

lx1 Px  (9) lx The last row  of the life table corresponds to the limit age, as typical values are 90, 100, 110 years.

Life tables - key parameters and relationships between them 473

Evaluation of the final coefficient L for the last year  is known as closing the life table. There are several suggestions for solving this problem – imputation, using a death rate, trend, etc. For the first birth year, the mortality rate is very high. For this reason, the coefficient before dx in formula (5) includes a selection of different values or other approaches [4]. The ratios (1)-(9) are associated with two adjacent ages. For longer periods, the ratios are shown below. From the equations

lx  lx2  dx  dx1 and lx  lx3  dx  dx1  dx2 follows that

lx lxn  dx  dx1  ...  dxn1 (10)

In the limit case formula (10) is

lx  dx  dx1  ... dm (11) which means that each person who reached the age of x will die between the age of x and the limit age. In short form the formulae (10) - (11) are xn1 n dx  lx  lxn  dt (12) tx  lx  dt (13) tx For longer periods, the following probability characteristics may be introduced:

 n px - probability for a person of the age of x to live another n years

lxn n px  (14) lx

 n qx - probability for a person of the age of x , to die over the next n years

lx  lxn n qx n qx 1n px   (15) lx lx

For the probability n px , the following ratios are fulfilled:

lx2 lx1 lx2 lx3 lx1 lx2 lx3 2 px   .  px .px1 , 3 px   . .  px .px1.px2 lx lx lx1 lx lx lx1 lx2 (16) Thus

lxn lx1 lx2 lxn n px   . ...  px .px1...pxn1 (17) lx lx lx1 lxn1 The life table may be drawn up also based on the introduced probabilistic characteristics (14)-(17). When using a hypothesis for even distribution of the mortality in the age interval, the number of person-years accumulated by the entire group in the interval x, x  n is

474 Ts. P. Kovacheva

l  l L  x xn n (18) n x 2 All the quantities in the life tables are treated as random variables. Therefore they are point estimates of theoretical functions [1].

3 Illustrative Example

Table 1 and Table 2 present parts of typical complete life tables, based on gender, for the population of Bulgaria (BG) and United States (USA) for 2012. The data for 2012 are taken, because such data are available for both countries. The tables contain data for every single year of age. The data in Table 1 are taken from the website of the National Statistical Institute of Bulgaria for 2011-2013 [3].

Table 1: Bulgarian life table

0 In this table are given only the values for qx , px and ex by age and gender of the chosen year. Thus, basing on the values for the parameter qx , the values of the 0 other parameters lx1 , d x , Lx , Tx and ex in this life table can be restored as follows. By the formulae (3) and (4) consequently are defined lx1 and dx with the base l0 100,000. By using formula (5) are calculated the values of Lx without the values for the first and the last year. Then, from formula (7) and (6) at 0 0 0  100 and given values е100 , е0 are determined T100  l100e100 , T100  L100 and

Life tables - key parameters and relationships between them 475

0 T0  l0e0 . The other values of Tx are calculated in the same way or by using formula (7). The first value of Lx is L0  Т0 Т1 . 0 The table values for the demographic parameters qx , px and ex by age x and gender for the selected year are given. Therefore, taking the values of the parameter qx and using the formulae (4) and (3), consequently are determined d x 0 and lx1 at a base l0 100,000. The other parameters Lx , Tx and ex are calculated respectively by the formulae (5) - (7). In Table 2, the data are taken from National Center of Health Statistics (NCHS) 0 of USA [5]. The values of the parameters d x , qx , Lx , Tx and ex are given by age and gender.

Table 2: USA life table

From both tables, the following basic data can be determined: ' ''  infant mortality dim for boys and dim for girls (from birth to age 1) determinate by formula

d0 dim  .1000‰ (19) l0 Starting from 100 000 at age x 1, in Bulgaria there are 99 116 boys and 99 369 girls who survived - 884 boys and 631 girls who died. The infant mortality in promiles is

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884 631 d'  .1000 8.84‰, d''  .1000  6.31‰ im 100 000 im 100 000 At the same age, in USA, there are 99 350 boys and 99 457 girls who survived, and 650 boys and 543 girls died. The infant mortality is smaller than in Bulgaria 650 543 d'  .1000 6.50‰, d''  .1000 5.43‰ im 100 000 im 100 000

0  average life expectancy ex - the first calculated value indicates that the males 0 in Bulgaria live on average e0  71.02 years and the females, about 7 years more - 0 0 to e0  78.01 years. In USA, the males live on average e0  76.42 years and the 0 females, about 5 years more – to e0  81.17 years. From Table 1, it can be determined than in Bulgaria, the retirement age of 63 years will be reached by 74 837 males and will predicatively live additional 15.2 years. Females reaching the age of 61 years will be 90 061 and will predicatively live additional 20.6 years. From Table 2, it can be determined that in USA, the retirement age of 65 will be reached by 80 724 males and 88 070 males who will predicatively live respectively additional 17.9 years and 20.5 years. Moreover, the life tables show that while in Bulgarian 189 males and 487 females will celebrate their 100th anniversary, in the USA they are 5 times more – 1 001 males and 2 784 females.

The variation of the number of people lx surviving to age x , for males and females in both countries is presented in Figure 1. Graphically, the distribution of the generation by gender and age for the Bulgaria and USA can be shown in the age-gender pyramid (Figure 3 and Figure 4). The diagrams consist of two back-to- back bar graphs, on which the population is plotted on the X-axis and age on the Y-axes. The left is showing the number of males and right – the number of females.

100 000 5000 males - males - USA USA 80 000 4000 females females - USA 60 000 - USA

3000 males - x

x males - l d BG 40 000 BG 2000 females females - BG 20 000 - BG 1000

0 0 20 40 60 80 100 0 0 20 40 60 80 100 x x

Figure 1: Number of surviving Figure 2: Number of deaths

Life tables - key parameters and relationships between them 477

From the Americans, about 73% of the males and 81% of the females will celebrate 70 years and from the Bulgarians - about 59% of the males and 79% of the females. The people who had lived to the age of 100 from the Americans are 466 males and 566 males, and from the Bulgarians - are less, 94 males and 305 females.

females - age females - age BG USA males - males - 90 BG 90 USA 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 100000 50000 0 50000 100000 100000 50000 0 50000 100000 lx lx

Figure 3: Age-gender pyramid (BG) Figure 4: Age-gender pyramid (USA)

The calculations based on the initial data from the considered life tables show that:  the quartile numbers of the people who lived between the initial aggregate l0 to:

- Q1  x0,25 =25% - for the Bulgarians: between 82-83 years for males and 87-88 for females and for the Americans: between 87-88 years for males and 91-92 years for females.

- Q2  x0,50 =50% - for the Bulgarians: between 73-74 years for males and 81-82 years for females and for the Americans: between 80-81 years for males and 84- 85 years for females;

- Q3  x0,75 =75% - for the Bulgarians: between 62-63 years for males and 73-74 years for females and for the Americans: between 69-70 years for males and 75- 76 years for females;

 number of people died in the considered interval - the data for dx indicate, as an absolute number, the decrease of the first main aggregate in its transition to the next age (Figure 2). The maximum of the mortality is:

- for the Bulgarians - d x  3 080 male at the age of x  78 years and d x  4 480 female at the age of x  84 years. - for the Americans is 3 461 male at the age of 86 years and 3 954 female at the age of 88 years;

- determination of the dying probabilities qx - the probabilities of dying (Figure 6) indicate what part of those who will survive to the age of x , will die within one year.

478 Ts. P. Kovacheva

Most deaths in Bulgaria are concentrated in the interval 77-(78)-81 years for males and the interval 81-(84)-88 years for females. Most deaths in the USA are concentrated in the interval 78-(86)-88 years for males and 82-(84)-93 years for females.

0,50 100,0 males - males - USA USA 0,40 80,0 females - females - USA USA 0,30

60,0 x

0 males - males -

x q

e BG BG 0,20 40,0 females - females - BG BG 20,0 0,10

0,0 0,00 0 20 40 60 80 100 0 20 40 60 80 100 x x

Figure 5: Average life expectancy Figure 6: Probability of dying

0 The results for the average life expectancy ex received for the groups considered is shown graphically in Figure 5. It is evident that the average life expectancy significantly decreases with increasing age. The ratio male-female is

qxm Qx  (20) qxf where - qxm , qxf - probabilities of dying for males and females between ages x and x 1.

In the Figure 7 the ratio Qx has been presented for Bulgaria and USA. For Bulgarians, the variations are greater. There are three peak values - 3.6 at 13 years, 3.2 years at 22 and 2.7 at 60 years. In the period 40-70 years mortality of males exceeds that of females by 2-2.7 times. Between 80 -100 years the ratio decreases to 1.

4,00 USA 3,50 3,00 BG 2,50 x 2,00 Q 1,50 1,00 0,50 0,00 0 20 40 60 80 100 x

Figure 7: Male-female ratio

Life tables - key parameters and relationships between them 479

For Americans, the ratio variations are smaller. The maximal values are 2.8 at 21 years and 1.7 at 59 years. Between 60-80 years it gradually decreases, and after 80 years decreases also to 1.

4 Conclusion

The life tables provide data to statisticians, demographers, insurers and specialists working in the field of social and pension insurance for: - amendment of the people number in a generation in the transition from age to age under the influence of the factor mortality; - analysis of the dynamics of mortality to characterize the mortality rate of the entire population or of separate age groups; - determination of the average life expectancy after a certain age; - determination of the influence of the mortality on the course of other demographic processes etc. According to the data from the life table of the conditional generation can be predicted the processes of development of the generations by gender and age for a given period of time, to be used then in different areas of the public life - increasing the average life expectancy and the retirement age etc.

References

[1] C. Chiang, Life Table and Mortality Analysis, World Health Organization, 1-413. http://apps.who.int/iris/bitstream/10665/62916/1/15736_eng.pdf

[2] D. Smith, N. Keyfitz, Mathematical Demography, Springer Verlag Berlin Heidelberg, 2013. https://doi.org/10.1007/978-3-642-35858-6

[3] Life tables, Republic of Bulgaria, National Statistical Institute. http://www.nsi.bg

[4] Life tables. http://papp.iussp.org/sessions/papp101_s07/PAPP101_s07_030_010.html

[5] National Center of Health Statistics (NCHS). https://www.cdc.gov/nchs/products/life_tables.htm

[6] N. Cholakov, On the importance of the social statistics for the development of the statistical method, 427-471. http://spaska.lirex.net/ins-educ/10.pdf

[7] R. Anderson, Method for Constructing Complete Annual U.S. Life Tables, Vital and Health Statistics, National Center of Health Statistics (NCHS), Series 2, no. 129, (1999), 1-36. https://www.cdc.gov/nchs/data/series/sr_02/sr02_129.pdf

Received: March 7, 2017; Published: March 21, 2017